Research ArticleSummation Formulas Involving Binomial CoefficientsHarmonic Numbers and Generalized Harmonic Numbers

Junesang Choi

Department of Mathematics Dongguk University Gyeongju 780-714 Republic of Korea

Correspondence should be addressed to Junesang Choi junesangmaildonggukackr

Received 19 May 2014 Accepted 30 June 2014 Published 21 July 2014

Academic Editor S D Purohit

Copyright copy 2014 Junesang ChoiThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distantpast and involved in a wide range of diverse fields such as analysis of algorithms in computer science various branches of numbertheory elementary particle physics and theoretical physics Here we show how one can obtain further interesting and (almost)serendipitous identities about certain finite or infinite series involving binomial coefficients harmonic numbers and generalizedharmonic numbers by simply applying the usual differential operator to well-known Gaussrsquos summation formula for

2F1(1)

1 Introduction and Preliminaries

The generalized harmonic numbers119867(119904)119899

of order 119904 which aredefined by (cf [1] see also [2 3] [4 page 156] and [5 Section35])

119867(119904)

119899=

119899

sum

119895=1

1

119895119904(119899 isin N 119904 isin C) (1)

119867119899= 119867(1)

119899=

119899

sum

119895=1

1

119895(119899 isin N) (2)

are the harmonic numbers Here N and C denote theset of positive integers and the set of complex numbersrespectively and we assume that

1198670= 0 119867

(119904)

0= 0 (119904 isin C 0)

119867(0)

0= 1

(3)

The generalized harmonic functions 119867(119904)119899(119911) are defined by

(see [2 6] see also [7 8])

119867(119904)

119899(119911) =

119899

sum

119895=1

1

(119895 + 119911)119904

(119899 isin N 119904 isin C Zminus

Zminus= minus1 minus2 minus3 )

(4)

so that obviously

119867(119904)

119899(0) = 119867

(119904)

119899 (5)

Equation (1) can be written in the following form

119867(119904)

119899= 120577 (119904) minus 120577 (119904 119899 + 1) (R (119904) gt 1 119899 isin N) (6)

by recalling the well-known (easily derivable) relationshipbetween the Riemann zeta function 120577(119904) and the Hurwitz (orgeneralized) zeta function 120577(119904 119886) (see [4 equation 23(9)])

120577 (119904) = 120577 (119904 119899 + 1) +

119899

sum

119896=1

119896minus119904

(119899 isin N0= N cup 0) (7)

The polygamma functions 120595(119899)(119904) (119899 isin N) are defined by

120595(119899)

(119904) =119889119899+1

119889119911119899+1log Γ (119904) = 119889

119899

119889119904119899120595 (119904)

(119899 isin N0 119904 isin C Z

minus

0= Zminuscup 0)

(8)

where Γ(119904) is the familiar gamma function and the psi-function 120595 is defined by

120595 (119904) =119889

119889119904log Γ (119904) 120595

(0)(119904) = 120595 (119904) (119904 isin C Z

minus

0) (9)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 501906 10 pageshttpdxdoiorg1011552014501906

2 Abstract and Applied Analysis

A well-known (and potentially useful) relationship be-tween the polygamma functions 120595(119899)(119904) and the generalizedzeta function 120577(119904 119886) is given by

120595(119899)

(119904) = (minus1)119899+1

119899

infin

sum

119896=0

1

(119896 + 119904)119899+1

= (minus1)119899+1

119899120577 (119899 + 1 119904)

(119899 isin N 119904 isin C Zminus

0)

(10)

It is also easy to have the following expression (cf [4 equation12(54)])

120595(119898)

(119904 + 119899) minus 120595(119898)

(119904) = (minus1)119898119898119867(119898+1)

119899(119904 minus 1)

(119898 119899 isin N0)

(11)

which immediately gives 119867(119904)119899

another expression for 119867(119904)119899

asfollows (cf [9 equation (20)])

119867(119898)

119899=

(minus1)119898minus1

(119898 minus 1)[120595(119898minus1)

(119899 + 1) minus 120595(119898minus1)

(1)]

(119898 isin N 119899 isin N0)

(12)

The following identitywas discovered byEuler in 1775 andhas a long history (see eg [10 page 252 et seq])

infin

sum

119899=1

119867119899

(119899 + 1)2=

1

2

infin

sum

119899=1

119867119899

1198992= 120577 (3) (13)

Identity (13) is a special case of the following more generalsum due to Euler

2

infin

sum

119896=1

119867119896

119896119899= (119899 + 2) 120577 (119899 + 1)

minus

119899minus2

sum

119896=1

120577 (119899 minus 119896) 120577 (119896 + 1)

(119899 isin N 1)

(14)

or equivalently

2

infin

sum

119896=1

119867119896

(119896 + 1)119899= 119899120577 (119899 + 1)

minus

119899minus2

sum

119896=1

120577 (119899 minus 119896) 120577 (119896 + 1)

(119899 isin N 1)

(15)

where (and in what follows) an empty sum is understood tobe nil

Many different techniques have been used in the vastmathematical literature in order to evaluate harmonic sumsof the types (13) and (15) For example D Borwein and JM Borwein [11] established the following interesting sums

by applying Parsevalrsquos identity to a Fourier series and contourintegrals to a generating function

infin

sum

119899=1

(119867119899

119899 + 1)

2

=11

4120577 (4) (16)

infin

sum

119899=1

(119867119899

119899)

2

=17

4120577 (4) (17)

infin

sum

119899=1

119867119899

1198993=

5

4120577 (4) (18)

where in light of Euler sum (18) nonlinear harmonic sums(16) and (17) are substantially the same since it is easilyverified that

infin

sum

119896=1

(119867119896

119896)

2

=

infin

sum

119896=1

(119867119896

119896 + 1)

2

+3

2120577 (4) (19)

Euler started this line of investigation in the course of hiscorrespondence with Goldbach beginning in 1742 and he wasthe first to consider the linear harmonic sums

S119901119902

=

infin

sum

119899=1

119867(119901)

119899

119899119902 (20)

Euler whose investigations were completed by Nielsenin 1906 (see Nielsen [12]) showed that the linear harmonicsums in (20) can be evaluated in terms of zeta values in thefollowing special cases 119901 = 1 119901 = 119902 119901 + 119902 odd and 119901 + 119902

even but with the pair (119901 119902) being restricted to a finite setof the so-called exceptional configurations (2 4) (4 2) (seeFlajolet and Salvy [13])Of these special cases in the oneswith119901 = 119902 if S

119901119902is known then S

119902119901can be found by means of

the symmetry relation

S119901119902

+S119902119901

= 120577 (119901) 120577 (119902) + 120577 (119901 + 119902) (21)

and vice versa (see also [1 page 140 Proposition 6] Sometypical examples are

infin

sum

119899=1

119867(2)

119899

1198992=

7

4120577 (4) (22)

infin

sum

119899=1

119867(2)

119899

1198995= 5120577 (2) 120577 (5) + 2120577 (3) 120577 (4) minus 10120577 (7) (23)

Rather extensive numerical search for linear relationsbetween linear Euler sums and polynomials in zeta values(see Bailey et al [14] see also Flajolet and Salvy [13]) stronglysuggests that Euler found all the possible evaluations of linearharmonic sums

The nonlinear harmonic sums involve products of at leasttwo harmonic numbers Let 119875 = (119901

1 119901

119896) be a partition of

an integer 119901 into 119896 summands so that 119901 = 1199011+ sdot sdot sdot + 119901

119896and

1199011≦ 1199012≦ sdot sdot sdot ≦ 119901

119896 The Euler sum of index 119875 119902 is defined by

S119875119902

=

infin

sum

119899=1

119867(1199011)

119899119867(1199012)

119899sdot sdot sdot 119867(119901119896)

119899

119899119902 (24)

Abstract and Applied Analysis 3

where the quantity 119902 + 1199011+ sdot sdot sdot + 119901

119896is called the weight

and the quantity 119896 is the degree A few basic nonlinearsums were recently evaluated by de Doelder [15] by invokingtheir relations with the Eulerian beta integrals or withpolylogarithm functions A detailed numerical search wasconducted by Bailey et al [14] who showed the existence ofmany surprising evaluations like

infin

sum

119899=1

(119867119899)3

1198994=

231

16120577 (7) minus

51

4120577 (3) 120577 (4) + 2120577 (2) 120577 (5) (25)

Flajolet and Salvy [13] clearly and extensively analyzedmost of the hitherto known evaluations for Euler sums andmultiple zeta functions (see also Hoffman [16] and Zagier[17]) For a remarkably clear and insightful exposition of sev-eral important results and conjectures concerning multiplepolylogarithms and the multiple zeta functions (includingespecially a broad survey of recent works on multiple zetaseries and Euler sums of arbitrary degree) the interestedreader should refer also to a survey-cum-expository paper byBowman and Bradley [18] which contains a fairly compre-hensive bibliography of as many as 83 further references onthe subject

Shen [19] investigated the connections between the Stir-ling numbers 119904(119899 119896) of the first kind and the Riemann zetafunction 120577(119899) bymeans of the Gauss summation formula (28)for the hypergeometric series In the course of his analysisShen [19] proved some known identities like (13) (18) and(22) In fact by employing the univariate series expansion ofclassical hypergeometric formulas Shen [19] and Choi andSrivastava [20 21] investigated the evaluation of infinite seriesrelated to generalized harmonic numbers On the other handmore summation formulas have been systematically derivedby Chu [22] and Chu and de Donno [23] who developed fullythis approach to the multivariate case We chose to recall twoidentities

infin

sum

119899=1

1198673

119899minus1minus 3119867119899minus1

119867(2)

119899minus1+ 2119867(3)

119899minus1

1198992= 6120577 (5)

infin

sum

119899=1

(119867(2)

119899

1198993+3119867119899

1198994) =

9

2120577 (5)

(26)

Many formulas of finite series involving binomial coef-ficients the Stirling numbers of the first and second kindsharmonic numbers and generalized harmonic numbers havealso been investigated in diverse ways (see eg [2 23ndash32])

Here we show how one can obtain further interestingand (almost) serendipitous identities about certain finiteor infinite series involving binomial coefficients harmonicnumbers and generalized harmonic numbers by simplyapplying the usual differential operator towell-knownGaussrsquos

summation formula for21198651(1) For example see the identi-

ties in Corollary 5

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3))

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(27)

Relevant connections between some of the identities pre-sented here with those in earlier works are also pointed out

2 Infinite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

We begin by recalling well-known Gaussrsquos summation for-mula for

21198651(1)

21198651(119886 119887 119888 1) =

infin

sum

119899=0

(119886)119899(119887)119899

119899(119888)119899

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

(R (119888 minus 119886 minus 119887) gt 0 119888 notin Zminus

0)

(28)

where (120572)119899denotes the Pochhammer symbol defined (for 120572 isin

C) by

(120572)119899=

1 (119899 = 0)

120572 (120572 + 1) sdot sdot sdot (120572 + 119899 minus 1) (119899 isin N) (29)

For convenient reference without proof we collect aset of easily derivable formulas necessary to provide furtherinteresting identities about certain finite or infinite seriesinvolving binomial coefficients harmonic numbers andgeneralized harmonic numbers asserted as in the followinglemma

Lemma 1 Each of the following identities holds true

119889

119889120572(120572)119899= (120572)119899119867(1)

119899(120572 minus 1) (119899 isin N

0 120572 isin C Z

minus

0)

119889

119889120572

1

(120572)119899

= minus119867(1)

119899(120572 minus 1)

(120572)119899

(119899 isin N0 120572 isin C Z

minus

0)

1198892

1198891205722(120572)119899= (120572)119899[119867(1)

119899(120572 minus 1)

2

minus 119867(2)

119899(120572 minus 1)]

(119899 isin N0 120572 isin C Z

minus

0)

4 Abstract and Applied Analysis

1198892

1198891205722

1

(120572)119899

=1

(120572)119899

[119867(1)

119899(120572 minus 1)

2

+ 119867(2)

119899(120572 minus 1)]

(119899 isin N0 120572 isin C Z

minus

0)

119889ℓ

119889119911ℓ119867(119904)

119899(119911) = (minus1)

ℓ(119904)ℓ119867(119904+ℓ)

119899(119911)

(119899 isin N ℓ isin N0 119904 isin C Z

minus)

119889

119889119886Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

119889

119889119888Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot (120595 (119888) + 120595 (119888 minus 119886 minus 119887)

minus120595 (119888 minus 119886) minus 120595 (119888 minus 119887))

(30)

Now we are ready to present certain general identitiesof infinite series involving binomial coefficients harmonicnumbers and generalized harmonic numbers as in thefollowing theorem

Theorem 2 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

(31)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [(120595(119888 minus 119886) minus 120595(119888 minus 119886 minus 119887))

2

minus (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))]

(32)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(2)

119899(119886 minus 1)

+ 2119867(3)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

3

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887)) ]

(33)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

119867(2)

119899(119886 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(3)

119899(119886 minus 1)

+3119867(2)

119899(119886 minus 1)

2

minus 6119867(4)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

4

minus 6(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))2

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 3(1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) minus 120595(3)

(119888 minus 119886 minus 119887)) ]

(34)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

+1205951015840(119888 minus 119886 minus 119887)]

(35)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

minus119867(1)

119899(119886 minus 1)119867

(2)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

+ 2 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(2)(119888 minus 119886 minus 119887)

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))]

(36)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(2)

119899(119887 minus 1)

+2119867(1)

119899(119886 minus 1)119867

(3)

119899(119887 minus 1) ]

Abstract and Applied Analysis 5

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

+ 3(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 1205951015840(119888 minus 119886 minus 119887) + 120595

(3)(119888 minus 119886 minus 119887)

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 3 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) minus 31205951015840(119888 minus 119886 minus 119887)

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))]

(37)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

4

minus 6119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

119867(2)

119899(119887 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(3)

119899(119887 minus 1)

+ 3119867(1)

119899(119886 minus 1) 119867

(2)

119899(119887 minus 1)

2

minus 6119867(1)

119899(119886 minus 1)119867

(4)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))4

+ 4(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(4)(119888 minus 119886 minus 119887)

minus 6 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 6(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 120595(2)

(119888 minus 119886 minus 119887)

minus 12 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(3)

(119888 minus 119886 minus 119887)

+ 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

2

+ 6 (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) + 4 1205951015840(119888 minus 119886 minus 119887)

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(3)

(119888 minus 119887) minus 120595(3)

(119888 minus 119886 minus 119887))]

(38)

Proof Differentiating each side of (28) with respect to thevariable 119886 and using some suitable identities in Section 1 andLemma 1 we obtain (31) Differentiating each side of (31) withrespect to the variable 119886 we get (32) Similarly we prove (33)and (34) Differentiating each side of (31) with respect to thevariable 119887 and using some suitable identities in Section 1 andLemma 1 we obtain (35) Similarly we prove (36) (37) and(38)

Setting 119888 = 2 and 119886 = 119887 = 1 in (31) to (34) and us-ing some suitable identities in Section 1 we obtain someinteresting identities involving harmonic numbers and gen-eralized harmonic numbers given in the following corollary

Corollary 3 Each of the following identities holds trueinfin

sum

119899=1

119867119899

(119899 + 1) (119899 + 2)= 1

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)2minus 119867(2)

119899] = 2 = 1 sdot 2

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)3minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

= 6 = 2 sdot 3

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)4

minus 6(119867119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

minus 6119867(4)

119899]

= 24 = 6 sdot 4

(39)

6 Abstract and Applied Analysis

Differentiating only the left-hand side of (34) with respectto the variable 119886 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (39) we may guess twoidentities asserted by the following conjecture

Conjecture 4 Each of the following identities may hold trueinfin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)5minus 10(119867

119899)3119867(2)

119899+ 20(119867

119899)2119867(3)

119899

+ 15119867119899(119867(2)

119899)2

minus 30119867119899119867(4)

119899

minus20119867(2)

119899119867(3)

119899+ 24119867

(5)

119899]

= 120 = 24 sdot 5

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6

minus 15(119867119899)4

119867(2)

119899+ 40(119867

119899)3

119867(3)

119899

+ 45(119867119899)2

(119867(2)

119899)2

minus 90(119867119899)2119867(4)

119899

minus 120119867119899119867(2)

119899119867(3)

119899+ 144119867

119899119867(5)

119899

minus 15(119867(2)

119899)3

+ 90119867(2)

119899119867(4)

119899+ 40(119867

(3)

119899)2

minus120119867(5)

119899] = 720 = 120 sdot 6

(40)

Setting 119888 = 3 and 119886 = 119887 = 1 in (35) to (44) and using somesuitable identities in Section 1 and Lemma 1 we obtain a setof very interesting identities in the following corollary

Corollary 5 Each of the following identities holds true

infin

sum

119899=1

(119867119899)2

(119899 + 1) (119899 + 2)= 2 (1 + 120577 (2)) (41)

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3)) (42)

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(43)

infin

sum

119899=1

(((119867119899)5minus 6(119867

119899)3119867(2)

119899+ 8(119867

119899)2

119867(3)

119899

+3119867119899(119867(2)

119899)2

minus 6119867119899119867(4)

119899)

times((119899 + 1) (119899 + 2))minus1)

= 48 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5))

(44)

Differentiating only the left-hand side of (38) with respectto the variable 119887 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (41) to (44) wemay guesstwo identities asserted by the following conjecture

Conjecture 6 Each of the following identities may hold true

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6minus 10(119867

119899)4119867(2)

119899+ 20(119867

119899)3119867(3)

119899

+ 15(119867119899)2(119867(2)

119899)2

minus 30(119867119899)2119867(4)

119899

minus20119867119899119867(2)

119899119867(3)

119899+ 24119867

119899119867(5)

119899]

= 240 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) + 120577 (6))

(45)infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)7minus 15(119867

119899)5119867(2)

119899+ 40(119867

119899)4119867(3)

119899

+ 45(119867119899)3(119867119899)2minus 90(119867

119899)3119867(4)

119899

minus 120(119867119899)2

119867(2)

119899119867(3)

119899+ 144(119867

119899)2

119867(5)

119899

minus 15119867119899(119867(2)

119899)3

+ 90119867119899119867(2)

119899119867(4)

119899

+40119867119899(119867(3)

119899)2

minus 120119867119899119867(6)

119899]

= 1440 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) +120577 (6) + 120577 (7))

(46)

Remark 7 It is interesting to observe that the number ofterms in the numerator of each of the left-hand sides of(41) to (46) is equal to the number of partitions of 119896 minus

1 (119896 = 2 3 4 5 6 7) respectively For example for (46) allthe partitions of 7 minus 1 = 6 are as follows

7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

= 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 2 + 2

= 1 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 3

= 1 + 1 + 2 + 3 = 1 + 3 + 3

= 1 + 1 + 1 + 4 = 1 + 1 + 5 = 1 + 2 + 4

(47)

where 119896 denotes119867(119896)119899

(119896 = 1 2 3 4 5) and + is translated intoa multiplication of its corresponding 119867

(119896)

119899 The coefficient of

each of the right-hand sides of (41) to (46) has the followingrule

2 4 = 2 sdot 2 12 = 4 sdot 3 48 = 12 sdot 4

240 = 48 sdot 5 1440 = 240 sdot 6

(48)

The remaining thing is to find a rule that dominates thecoefficients of each term of the numerator of the left-handsides of (41) to (46)

Differentiating each side of (28) with respect to thevariable 119888 successively and using some suitable identitiesin Section 1 and Lemma 1 we obtain a set of infinite

Abstract and Applied Analysis 7

series involving binomial coefficients harmonic numbersand generalized harmonic numbers different from those inTheorem 2 as in the following theorem

Theorem 8 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119888 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

minus (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

3

+ 3119867(1)

119899(119888 minus 1)

times119867(2)

119899(119888 minus 1) + 2119867

(3)

119899(119888 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))3

minus 3 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887)

minus 120595(2)

(119888) minus 120595(2)

(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

4

+ 6119867(1)

119899(119888 minus 1)

2

times 119867(2)

119899(119888 minus 1) + 8119867

(1)

119899(119888 minus 1)119867

(3)

119899(119888 minus 1)

+ 3119867(2)

119899(119888 minus 1)

2

+ 6119867(4)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))4

minus 6(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887) minus 120595(2)

(119888)

minus120595(2)

(119888 minus 119886 minus 119887))

+ 3 (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888)

minus1205951015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) + 120595(3)

(119888 minus 119887) minus 120595(3)

(119888)

minus120595(3)

(119888 minus 119886 minus 119887)) ]

(49)

Setting 119888 = 1 and 119886 = 119887 = minus12 in (49) and usingsome suitable identities in Section 1 and special values of 120595-function (see eg [4 Section 12] and [5 Section 13]) weobtain a set of interesting infinite series involving binomialcoefficients and harmonic numbers given in the followingcorollary

Corollary 9 Each of the following identities holds true

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

119867119899=

4

120587(3 minus 4 log 2)

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)2+ 119867(2)

119899]

=4

120587[(3 minus 4 log 2)2 + 7 minus 4120577 (2)]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)3+ 3119867119899119867(2)

119899+ 2119867(3)

119899]

=4

120587[(3 minus 4 log 2)3 minus 3 (3 minus 4 log 2) (4120577 (2) minus 7)

minus24120577 (3) + 30]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)4+ 6(119867

119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

+ 6119867(4)

119899]

=4

120587[(3 minus 4 log 2)4 minus 6(3 minus 4 log 2)2 (4120577 (2) minus 7)

+ 24 (3 minus 4 log 2) (5 minus 4120577 (3)) + 3(4120577 (2) minus 7)2

+186 minus 168120577 (4) ]

(50)

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

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Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

A well-known (and potentially useful) relationship be-tween the polygamma functions 120595(119899)(119904) and the generalizedzeta function 120577(119904 119886) is given by

120595(119899)

(119904) = (minus1)119899+1

119899

infin

sum

119896=0

1

(119896 + 119904)119899+1

= (minus1)119899+1

119899120577 (119899 + 1 119904)

(119899 isin N 119904 isin C Zminus

0)

(10)

It is also easy to have the following expression (cf [4 equation12(54)])

120595(119898)

(119904 + 119899) minus 120595(119898)

(119904) = (minus1)119898119898119867(119898+1)

119899(119904 minus 1)

(119898 119899 isin N0)

(11)

which immediately gives 119867(119904)119899

another expression for 119867(119904)119899

asfollows (cf [9 equation (20)])

119867(119898)

119899=

(minus1)119898minus1

(119898 minus 1)[120595(119898minus1)

(119899 + 1) minus 120595(119898minus1)

(1)]

(119898 isin N 119899 isin N0)

(12)

The following identitywas discovered byEuler in 1775 andhas a long history (see eg [10 page 252 et seq])

infin

sum

119899=1

119867119899

(119899 + 1)2=

1

2

infin

sum

119899=1

119867119899

1198992= 120577 (3) (13)

Identity (13) is a special case of the following more generalsum due to Euler

2

infin

sum

119896=1

119867119896

119896119899= (119899 + 2) 120577 (119899 + 1)

minus

119899minus2

sum

119896=1

120577 (119899 minus 119896) 120577 (119896 + 1)

(119899 isin N 1)

(14)

or equivalently

2

infin

sum

119896=1

119867119896

(119896 + 1)119899= 119899120577 (119899 + 1)

minus

119899minus2

sum

119896=1

120577 (119899 minus 119896) 120577 (119896 + 1)

(119899 isin N 1)

(15)

where (and in what follows) an empty sum is understood tobe nil

Many different techniques have been used in the vastmathematical literature in order to evaluate harmonic sumsof the types (13) and (15) For example D Borwein and JM Borwein [11] established the following interesting sums

by applying Parsevalrsquos identity to a Fourier series and contourintegrals to a generating function

infin

sum

119899=1

(119867119899

119899 + 1)

2

=11

4120577 (4) (16)

infin

sum

119899=1

(119867119899

119899)

2

=17

4120577 (4) (17)

infin

sum

119899=1

119867119899

1198993=

5

4120577 (4) (18)

where in light of Euler sum (18) nonlinear harmonic sums(16) and (17) are substantially the same since it is easilyverified that

infin

sum

119896=1

(119867119896

119896)

2

=

infin

sum

119896=1

(119867119896

119896 + 1)

2

+3

2120577 (4) (19)

Euler started this line of investigation in the course of hiscorrespondence with Goldbach beginning in 1742 and he wasthe first to consider the linear harmonic sums

S119901119902

=

infin

sum

119899=1

119867(119901)

119899

119899119902 (20)

Euler whose investigations were completed by Nielsenin 1906 (see Nielsen [12]) showed that the linear harmonicsums in (20) can be evaluated in terms of zeta values in thefollowing special cases 119901 = 1 119901 = 119902 119901 + 119902 odd and 119901 + 119902

even but with the pair (119901 119902) being restricted to a finite setof the so-called exceptional configurations (2 4) (4 2) (seeFlajolet and Salvy [13])Of these special cases in the oneswith119901 = 119902 if S

119901119902is known then S

119902119901can be found by means of

the symmetry relation

S119901119902

+S119902119901

= 120577 (119901) 120577 (119902) + 120577 (119901 + 119902) (21)

and vice versa (see also [1 page 140 Proposition 6] Sometypical examples are

infin

sum

119899=1

119867(2)

119899

1198992=

7

4120577 (4) (22)

infin

sum

119899=1

119867(2)

119899

1198995= 5120577 (2) 120577 (5) + 2120577 (3) 120577 (4) minus 10120577 (7) (23)

Rather extensive numerical search for linear relationsbetween linear Euler sums and polynomials in zeta values(see Bailey et al [14] see also Flajolet and Salvy [13]) stronglysuggests that Euler found all the possible evaluations of linearharmonic sums

The nonlinear harmonic sums involve products of at leasttwo harmonic numbers Let 119875 = (119901

1 119901

119896) be a partition of

an integer 119901 into 119896 summands so that 119901 = 1199011+ sdot sdot sdot + 119901

119896and

1199011≦ 1199012≦ sdot sdot sdot ≦ 119901

119896 The Euler sum of index 119875 119902 is defined by

S119875119902

=

infin

sum

119899=1

119867(1199011)

119899119867(1199012)

119899sdot sdot sdot 119867(119901119896)

119899

119899119902 (24)

Abstract and Applied Analysis 3

where the quantity 119902 + 1199011+ sdot sdot sdot + 119901

119896is called the weight

and the quantity 119896 is the degree A few basic nonlinearsums were recently evaluated by de Doelder [15] by invokingtheir relations with the Eulerian beta integrals or withpolylogarithm functions A detailed numerical search wasconducted by Bailey et al [14] who showed the existence ofmany surprising evaluations like

infin

sum

119899=1

(119867119899)3

1198994=

231

16120577 (7) minus

51

4120577 (3) 120577 (4) + 2120577 (2) 120577 (5) (25)

Flajolet and Salvy [13] clearly and extensively analyzedmost of the hitherto known evaluations for Euler sums andmultiple zeta functions (see also Hoffman [16] and Zagier[17]) For a remarkably clear and insightful exposition of sev-eral important results and conjectures concerning multiplepolylogarithms and the multiple zeta functions (includingespecially a broad survey of recent works on multiple zetaseries and Euler sums of arbitrary degree) the interestedreader should refer also to a survey-cum-expository paper byBowman and Bradley [18] which contains a fairly compre-hensive bibliography of as many as 83 further references onthe subject

Shen [19] investigated the connections between the Stir-ling numbers 119904(119899 119896) of the first kind and the Riemann zetafunction 120577(119899) bymeans of the Gauss summation formula (28)for the hypergeometric series In the course of his analysisShen [19] proved some known identities like (13) (18) and(22) In fact by employing the univariate series expansion ofclassical hypergeometric formulas Shen [19] and Choi andSrivastava [20 21] investigated the evaluation of infinite seriesrelated to generalized harmonic numbers On the other handmore summation formulas have been systematically derivedby Chu [22] and Chu and de Donno [23] who developed fullythis approach to the multivariate case We chose to recall twoidentities

infin

sum

119899=1

1198673

119899minus1minus 3119867119899minus1

119867(2)

119899minus1+ 2119867(3)

119899minus1

1198992= 6120577 (5)

infin

sum

119899=1

(119867(2)

119899

1198993+3119867119899

1198994) =

9

2120577 (5)

(26)

Many formulas of finite series involving binomial coef-ficients the Stirling numbers of the first and second kindsharmonic numbers and generalized harmonic numbers havealso been investigated in diverse ways (see eg [2 23ndash32])

Here we show how one can obtain further interestingand (almost) serendipitous identities about certain finiteor infinite series involving binomial coefficients harmonicnumbers and generalized harmonic numbers by simplyapplying the usual differential operator towell-knownGaussrsquos

summation formula for21198651(1) For example see the identi-

ties in Corollary 5

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3))

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(27)

Relevant connections between some of the identities pre-sented here with those in earlier works are also pointed out

2 Infinite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

We begin by recalling well-known Gaussrsquos summation for-mula for

21198651(1)

21198651(119886 119887 119888 1) =

infin

sum

119899=0

(119886)119899(119887)119899

119899(119888)119899

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

(R (119888 minus 119886 minus 119887) gt 0 119888 notin Zminus

0)

(28)

where (120572)119899denotes the Pochhammer symbol defined (for 120572 isin

C) by

(120572)119899=

1 (119899 = 0)

120572 (120572 + 1) sdot sdot sdot (120572 + 119899 minus 1) (119899 isin N) (29)

For convenient reference without proof we collect aset of easily derivable formulas necessary to provide furtherinteresting identities about certain finite or infinite seriesinvolving binomial coefficients harmonic numbers andgeneralized harmonic numbers asserted as in the followinglemma

Lemma 1 Each of the following identities holds true

119889

119889120572(120572)119899= (120572)119899119867(1)

119899(120572 minus 1) (119899 isin N

0 120572 isin C Z

minus

0)

119889

119889120572

1

(120572)119899

= minus119867(1)

119899(120572 minus 1)

(120572)119899

(119899 isin N0 120572 isin C Z

minus

0)

1198892

1198891205722(120572)119899= (120572)119899[119867(1)

119899(120572 minus 1)

2

minus 119867(2)

119899(120572 minus 1)]

(119899 isin N0 120572 isin C Z

minus

0)

4 Abstract and Applied Analysis

1198892

1198891205722

1

(120572)119899

=1

(120572)119899

[119867(1)

119899(120572 minus 1)

2

+ 119867(2)

119899(120572 minus 1)]

(119899 isin N0 120572 isin C Z

minus

0)

119889ℓ

119889119911ℓ119867(119904)

119899(119911) = (minus1)

ℓ(119904)ℓ119867(119904+ℓ)

119899(119911)

(119899 isin N ℓ isin N0 119904 isin C Z

minus)

119889

119889119886Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

119889

119889119888Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot (120595 (119888) + 120595 (119888 minus 119886 minus 119887)

minus120595 (119888 minus 119886) minus 120595 (119888 minus 119887))

(30)

Now we are ready to present certain general identitiesof infinite series involving binomial coefficients harmonicnumbers and generalized harmonic numbers as in thefollowing theorem

Theorem 2 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

(31)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [(120595(119888 minus 119886) minus 120595(119888 minus 119886 minus 119887))

2

minus (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))]

(32)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(2)

119899(119886 minus 1)

+ 2119867(3)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

3

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887)) ]

(33)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

119867(2)

119899(119886 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(3)

119899(119886 minus 1)

+3119867(2)

119899(119886 minus 1)

2

minus 6119867(4)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

4

minus 6(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))2

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 3(1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) minus 120595(3)

(119888 minus 119886 minus 119887)) ]

(34)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

+1205951015840(119888 minus 119886 minus 119887)]

(35)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

minus119867(1)

119899(119886 minus 1)119867

(2)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

+ 2 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(2)(119888 minus 119886 minus 119887)

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))]

(36)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(2)

119899(119887 minus 1)

+2119867(1)

119899(119886 minus 1)119867

(3)

119899(119887 minus 1) ]

Abstract and Applied Analysis 5

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

+ 3(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 1205951015840(119888 minus 119886 minus 119887) + 120595

(3)(119888 minus 119886 minus 119887)

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 3 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) minus 31205951015840(119888 minus 119886 minus 119887)

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))]

(37)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

4

minus 6119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

119867(2)

119899(119887 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(3)

119899(119887 minus 1)

+ 3119867(1)

119899(119886 minus 1) 119867

(2)

119899(119887 minus 1)

2

minus 6119867(1)

119899(119886 minus 1)119867

(4)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))4

+ 4(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(4)(119888 minus 119886 minus 119887)

minus 6 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 6(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 120595(2)

(119888 minus 119886 minus 119887)

minus 12 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(3)

(119888 minus 119886 minus 119887)

+ 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

2

+ 6 (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) + 4 1205951015840(119888 minus 119886 minus 119887)

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(3)

(119888 minus 119887) minus 120595(3)

(119888 minus 119886 minus 119887))]

(38)

Proof Differentiating each side of (28) with respect to thevariable 119886 and using some suitable identities in Section 1 andLemma 1 we obtain (31) Differentiating each side of (31) withrespect to the variable 119886 we get (32) Similarly we prove (33)and (34) Differentiating each side of (31) with respect to thevariable 119887 and using some suitable identities in Section 1 andLemma 1 we obtain (35) Similarly we prove (36) (37) and(38)

Setting 119888 = 2 and 119886 = 119887 = 1 in (31) to (34) and us-ing some suitable identities in Section 1 we obtain someinteresting identities involving harmonic numbers and gen-eralized harmonic numbers given in the following corollary

Corollary 3 Each of the following identities holds trueinfin

sum

119899=1

119867119899

(119899 + 1) (119899 + 2)= 1

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)2minus 119867(2)

119899] = 2 = 1 sdot 2

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)3minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

= 6 = 2 sdot 3

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)4

minus 6(119867119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

minus 6119867(4)

119899]

= 24 = 6 sdot 4

(39)

6 Abstract and Applied Analysis

Differentiating only the left-hand side of (34) with respectto the variable 119886 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (39) we may guess twoidentities asserted by the following conjecture

Conjecture 4 Each of the following identities may hold trueinfin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)5minus 10(119867

119899)3119867(2)

119899+ 20(119867

119899)2119867(3)

119899

+ 15119867119899(119867(2)

119899)2

minus 30119867119899119867(4)

119899

minus20119867(2)

119899119867(3)

119899+ 24119867

(5)

119899]

= 120 = 24 sdot 5

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6

minus 15(119867119899)4

119867(2)

119899+ 40(119867

119899)3

119867(3)

119899

+ 45(119867119899)2

(119867(2)

119899)2

minus 90(119867119899)2119867(4)

119899

minus 120119867119899119867(2)

119899119867(3)

119899+ 144119867

119899119867(5)

119899

minus 15(119867(2)

119899)3

+ 90119867(2)

119899119867(4)

119899+ 40(119867

(3)

119899)2

minus120119867(5)

119899] = 720 = 120 sdot 6

(40)

Setting 119888 = 3 and 119886 = 119887 = 1 in (35) to (44) and using somesuitable identities in Section 1 and Lemma 1 we obtain a setof very interesting identities in the following corollary

Corollary 5 Each of the following identities holds true

infin

sum

119899=1

(119867119899)2

(119899 + 1) (119899 + 2)= 2 (1 + 120577 (2)) (41)

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3)) (42)

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(43)

infin

sum

119899=1

(((119867119899)5minus 6(119867

119899)3119867(2)

119899+ 8(119867

119899)2

119867(3)

119899

+3119867119899(119867(2)

119899)2

minus 6119867119899119867(4)

119899)

times((119899 + 1) (119899 + 2))minus1)

= 48 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5))

(44)

Differentiating only the left-hand side of (38) with respectto the variable 119887 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (41) to (44) wemay guesstwo identities asserted by the following conjecture

Conjecture 6 Each of the following identities may hold true

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6minus 10(119867

119899)4119867(2)

119899+ 20(119867

119899)3119867(3)

119899

+ 15(119867119899)2(119867(2)

119899)2

minus 30(119867119899)2119867(4)

119899

minus20119867119899119867(2)

119899119867(3)

119899+ 24119867

119899119867(5)

119899]

= 240 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) + 120577 (6))

(45)infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)7minus 15(119867

119899)5119867(2)

119899+ 40(119867

119899)4119867(3)

119899

+ 45(119867119899)3(119867119899)2minus 90(119867

119899)3119867(4)

119899

minus 120(119867119899)2

119867(2)

119899119867(3)

119899+ 144(119867

119899)2

119867(5)

119899

minus 15119867119899(119867(2)

119899)3

+ 90119867119899119867(2)

119899119867(4)

119899

+40119867119899(119867(3)

119899)2

minus 120119867119899119867(6)

119899]

= 1440 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) +120577 (6) + 120577 (7))

(46)

Remark 7 It is interesting to observe that the number ofterms in the numerator of each of the left-hand sides of(41) to (46) is equal to the number of partitions of 119896 minus

1 (119896 = 2 3 4 5 6 7) respectively For example for (46) allthe partitions of 7 minus 1 = 6 are as follows

7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

= 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 2 + 2

= 1 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 3

= 1 + 1 + 2 + 3 = 1 + 3 + 3

= 1 + 1 + 1 + 4 = 1 + 1 + 5 = 1 + 2 + 4

(47)

where 119896 denotes119867(119896)119899

(119896 = 1 2 3 4 5) and + is translated intoa multiplication of its corresponding 119867

(119896)

119899 The coefficient of

each of the right-hand sides of (41) to (46) has the followingrule

2 4 = 2 sdot 2 12 = 4 sdot 3 48 = 12 sdot 4

240 = 48 sdot 5 1440 = 240 sdot 6

(48)

The remaining thing is to find a rule that dominates thecoefficients of each term of the numerator of the left-handsides of (41) to (46)

Differentiating each side of (28) with respect to thevariable 119888 successively and using some suitable identitiesin Section 1 and Lemma 1 we obtain a set of infinite

Abstract and Applied Analysis 7

series involving binomial coefficients harmonic numbersand generalized harmonic numbers different from those inTheorem 2 as in the following theorem

Theorem 8 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119888 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

minus (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

3

+ 3119867(1)

119899(119888 minus 1)

times119867(2)

119899(119888 minus 1) + 2119867

(3)

119899(119888 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))3

minus 3 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887)

minus 120595(2)

(119888) minus 120595(2)

(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

4

+ 6119867(1)

119899(119888 minus 1)

2

times 119867(2)

119899(119888 minus 1) + 8119867

(1)

119899(119888 minus 1)119867

(3)

119899(119888 minus 1)

+ 3119867(2)

119899(119888 minus 1)

2

+ 6119867(4)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))4

minus 6(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887) minus 120595(2)

(119888)

minus120595(2)

(119888 minus 119886 minus 119887))

+ 3 (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888)

minus1205951015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) + 120595(3)

(119888 minus 119887) minus 120595(3)

(119888)

minus120595(3)

(119888 minus 119886 minus 119887)) ]

(49)

Setting 119888 = 1 and 119886 = 119887 = minus12 in (49) and usingsome suitable identities in Section 1 and special values of 120595-function (see eg [4 Section 12] and [5 Section 13]) weobtain a set of interesting infinite series involving binomialcoefficients and harmonic numbers given in the followingcorollary

Corollary 9 Each of the following identities holds true

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

119867119899=

4

120587(3 minus 4 log 2)

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)2+ 119867(2)

119899]

=4

120587[(3 minus 4 log 2)2 + 7 minus 4120577 (2)]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)3+ 3119867119899119867(2)

119899+ 2119867(3)

119899]

=4

120587[(3 minus 4 log 2)3 minus 3 (3 minus 4 log 2) (4120577 (2) minus 7)

minus24120577 (3) + 30]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)4+ 6(119867

119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

+ 6119867(4)

119899]

=4

120587[(3 minus 4 log 2)4 minus 6(3 minus 4 log 2)2 (4120577 (2) minus 7)

+ 24 (3 minus 4 log 2) (5 minus 4120577 (3)) + 3(4120577 (2) minus 7)2

+186 minus 168120577 (4) ]

(50)

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

where the quantity 119902 + 1199011+ sdot sdot sdot + 119901

119896is called the weight

and the quantity 119896 is the degree A few basic nonlinearsums were recently evaluated by de Doelder [15] by invokingtheir relations with the Eulerian beta integrals or withpolylogarithm functions A detailed numerical search wasconducted by Bailey et al [14] who showed the existence ofmany surprising evaluations like

infin

sum

119899=1

(119867119899)3

1198994=

231

16120577 (7) minus

51

4120577 (3) 120577 (4) + 2120577 (2) 120577 (5) (25)

Flajolet and Salvy [13] clearly and extensively analyzedmost of the hitherto known evaluations for Euler sums andmultiple zeta functions (see also Hoffman [16] and Zagier[17]) For a remarkably clear and insightful exposition of sev-eral important results and conjectures concerning multiplepolylogarithms and the multiple zeta functions (includingespecially a broad survey of recent works on multiple zetaseries and Euler sums of arbitrary degree) the interestedreader should refer also to a survey-cum-expository paper byBowman and Bradley [18] which contains a fairly compre-hensive bibliography of as many as 83 further references onthe subject

Shen [19] investigated the connections between the Stir-ling numbers 119904(119899 119896) of the first kind and the Riemann zetafunction 120577(119899) bymeans of the Gauss summation formula (28)for the hypergeometric series In the course of his analysisShen [19] proved some known identities like (13) (18) and(22) In fact by employing the univariate series expansion ofclassical hypergeometric formulas Shen [19] and Choi andSrivastava [20 21] investigated the evaluation of infinite seriesrelated to generalized harmonic numbers On the other handmore summation formulas have been systematically derivedby Chu [22] and Chu and de Donno [23] who developed fullythis approach to the multivariate case We chose to recall twoidentities

infin

sum

119899=1

1198673

119899minus1minus 3119867119899minus1

119867(2)

119899minus1+ 2119867(3)

119899minus1

1198992= 6120577 (5)

infin

sum

119899=1

(119867(2)

119899

1198993+3119867119899

1198994) =

9

2120577 (5)

(26)

Many formulas of finite series involving binomial coef-ficients the Stirling numbers of the first and second kindsharmonic numbers and generalized harmonic numbers havealso been investigated in diverse ways (see eg [2 23ndash32])

Here we show how one can obtain further interestingand (almost) serendipitous identities about certain finiteor infinite series involving binomial coefficients harmonicnumbers and generalized harmonic numbers by simplyapplying the usual differential operator towell-knownGaussrsquos

summation formula for21198651(1) For example see the identi-

ties in Corollary 5

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3))

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(27)

Relevant connections between some of the identities pre-sented here with those in earlier works are also pointed out

2 Infinite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

We begin by recalling well-known Gaussrsquos summation for-mula for

21198651(1)

21198651(119886 119887 119888 1) =

infin

sum

119899=0

(119886)119899(119887)119899

119899(119888)119899

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

(R (119888 minus 119886 minus 119887) gt 0 119888 notin Zminus

0)

(28)

where (120572)119899denotes the Pochhammer symbol defined (for 120572 isin

C) by

(120572)119899=

1 (119899 = 0)

120572 (120572 + 1) sdot sdot sdot (120572 + 119899 minus 1) (119899 isin N) (29)

For convenient reference without proof we collect aset of easily derivable formulas necessary to provide furtherinteresting identities about certain finite or infinite seriesinvolving binomial coefficients harmonic numbers andgeneralized harmonic numbers asserted as in the followinglemma

Lemma 1 Each of the following identities holds true

119889

119889120572(120572)119899= (120572)119899119867(1)

119899(120572 minus 1) (119899 isin N

0 120572 isin C Z

minus

0)

119889

119889120572

1

(120572)119899

= minus119867(1)

119899(120572 minus 1)

(120572)119899

(119899 isin N0 120572 isin C Z

minus

0)

1198892

1198891205722(120572)119899= (120572)119899[119867(1)

119899(120572 minus 1)

2

minus 119867(2)

119899(120572 minus 1)]

(119899 isin N0 120572 isin C Z

minus

0)

4 Abstract and Applied Analysis

1198892

1198891205722

1

(120572)119899

=1

(120572)119899

[119867(1)

119899(120572 minus 1)

2

+ 119867(2)

119899(120572 minus 1)]

(119899 isin N0 120572 isin C Z

minus

0)

119889ℓ

119889119911ℓ119867(119904)

119899(119911) = (minus1)

ℓ(119904)ℓ119867(119904+ℓ)

119899(119911)

(119899 isin N ℓ isin N0 119904 isin C Z

minus)

119889

119889119886Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

119889

119889119888Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot (120595 (119888) + 120595 (119888 minus 119886 minus 119887)

minus120595 (119888 minus 119886) minus 120595 (119888 minus 119887))

(30)

Now we are ready to present certain general identitiesof infinite series involving binomial coefficients harmonicnumbers and generalized harmonic numbers as in thefollowing theorem

Theorem 2 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

(31)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [(120595(119888 minus 119886) minus 120595(119888 minus 119886 minus 119887))

2

minus (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))]

(32)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(2)

119899(119886 minus 1)

+ 2119867(3)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

3

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887)) ]

(33)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

119867(2)

119899(119886 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(3)

119899(119886 minus 1)

+3119867(2)

119899(119886 minus 1)

2

minus 6119867(4)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

4

minus 6(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))2

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 3(1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) minus 120595(3)

(119888 minus 119886 minus 119887)) ]

(34)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

+1205951015840(119888 minus 119886 minus 119887)]

(35)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

minus119867(1)

119899(119886 minus 1)119867

(2)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

+ 2 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(2)(119888 minus 119886 minus 119887)

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))]

(36)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(2)

119899(119887 minus 1)

+2119867(1)

119899(119886 minus 1)119867

(3)

119899(119887 minus 1) ]

Abstract and Applied Analysis 5

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

+ 3(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 1205951015840(119888 minus 119886 minus 119887) + 120595

(3)(119888 minus 119886 minus 119887)

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 3 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) minus 31205951015840(119888 minus 119886 minus 119887)

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))]

(37)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

4

minus 6119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

119867(2)

119899(119887 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(3)

119899(119887 minus 1)

+ 3119867(1)

119899(119886 minus 1) 119867

(2)

119899(119887 minus 1)

2

minus 6119867(1)

119899(119886 minus 1)119867

(4)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))4

+ 4(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(4)(119888 minus 119886 minus 119887)

minus 6 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 6(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 120595(2)

(119888 minus 119886 minus 119887)

minus 12 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(3)

(119888 minus 119886 minus 119887)

+ 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

2

+ 6 (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) + 4 1205951015840(119888 minus 119886 minus 119887)

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(3)

(119888 minus 119887) minus 120595(3)

(119888 minus 119886 minus 119887))]

(38)

Proof Differentiating each side of (28) with respect to thevariable 119886 and using some suitable identities in Section 1 andLemma 1 we obtain (31) Differentiating each side of (31) withrespect to the variable 119886 we get (32) Similarly we prove (33)and (34) Differentiating each side of (31) with respect to thevariable 119887 and using some suitable identities in Section 1 andLemma 1 we obtain (35) Similarly we prove (36) (37) and(38)

Setting 119888 = 2 and 119886 = 119887 = 1 in (31) to (34) and us-ing some suitable identities in Section 1 we obtain someinteresting identities involving harmonic numbers and gen-eralized harmonic numbers given in the following corollary

Corollary 3 Each of the following identities holds trueinfin

sum

119899=1

119867119899

(119899 + 1) (119899 + 2)= 1

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)2minus 119867(2)

119899] = 2 = 1 sdot 2

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)3minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

= 6 = 2 sdot 3

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)4

minus 6(119867119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

minus 6119867(4)

119899]

= 24 = 6 sdot 4

(39)

6 Abstract and Applied Analysis

Differentiating only the left-hand side of (34) with respectto the variable 119886 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (39) we may guess twoidentities asserted by the following conjecture

Conjecture 4 Each of the following identities may hold trueinfin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)5minus 10(119867

119899)3119867(2)

119899+ 20(119867

119899)2119867(3)

119899

+ 15119867119899(119867(2)

119899)2

minus 30119867119899119867(4)

119899

minus20119867(2)

119899119867(3)

119899+ 24119867

(5)

119899]

= 120 = 24 sdot 5

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6

minus 15(119867119899)4

119867(2)

119899+ 40(119867

119899)3

119867(3)

119899

+ 45(119867119899)2

(119867(2)

119899)2

minus 90(119867119899)2119867(4)

119899

minus 120119867119899119867(2)

119899119867(3)

119899+ 144119867

119899119867(5)

119899

minus 15(119867(2)

119899)3

+ 90119867(2)

119899119867(4)

119899+ 40(119867

(3)

119899)2

minus120119867(5)

119899] = 720 = 120 sdot 6

(40)

Setting 119888 = 3 and 119886 = 119887 = 1 in (35) to (44) and using somesuitable identities in Section 1 and Lemma 1 we obtain a setof very interesting identities in the following corollary

Corollary 5 Each of the following identities holds true

infin

sum

119899=1

(119867119899)2

(119899 + 1) (119899 + 2)= 2 (1 + 120577 (2)) (41)

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3)) (42)

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(43)

infin

sum

119899=1

(((119867119899)5minus 6(119867

119899)3119867(2)

119899+ 8(119867

119899)2

119867(3)

119899

+3119867119899(119867(2)

119899)2

minus 6119867119899119867(4)

119899)

times((119899 + 1) (119899 + 2))minus1)

= 48 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5))

(44)

Differentiating only the left-hand side of (38) with respectto the variable 119887 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (41) to (44) wemay guesstwo identities asserted by the following conjecture

Conjecture 6 Each of the following identities may hold true

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6minus 10(119867

119899)4119867(2)

119899+ 20(119867

119899)3119867(3)

119899

+ 15(119867119899)2(119867(2)

119899)2

minus 30(119867119899)2119867(4)

119899

minus20119867119899119867(2)

119899119867(3)

119899+ 24119867

119899119867(5)

119899]

= 240 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) + 120577 (6))

(45)infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)7minus 15(119867

119899)5119867(2)

119899+ 40(119867

119899)4119867(3)

119899

+ 45(119867119899)3(119867119899)2minus 90(119867

119899)3119867(4)

119899

minus 120(119867119899)2

119867(2)

119899119867(3)

119899+ 144(119867

119899)2

119867(5)

119899

minus 15119867119899(119867(2)

119899)3

+ 90119867119899119867(2)

119899119867(4)

119899

+40119867119899(119867(3)

119899)2

minus 120119867119899119867(6)

119899]

= 1440 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) +120577 (6) + 120577 (7))

(46)

Remark 7 It is interesting to observe that the number ofterms in the numerator of each of the left-hand sides of(41) to (46) is equal to the number of partitions of 119896 minus

1 (119896 = 2 3 4 5 6 7) respectively For example for (46) allthe partitions of 7 minus 1 = 6 are as follows

7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

= 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 2 + 2

= 1 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 3

= 1 + 1 + 2 + 3 = 1 + 3 + 3

= 1 + 1 + 1 + 4 = 1 + 1 + 5 = 1 + 2 + 4

(47)

where 119896 denotes119867(119896)119899

(119896 = 1 2 3 4 5) and + is translated intoa multiplication of its corresponding 119867

(119896)

119899 The coefficient of

each of the right-hand sides of (41) to (46) has the followingrule

2 4 = 2 sdot 2 12 = 4 sdot 3 48 = 12 sdot 4

240 = 48 sdot 5 1440 = 240 sdot 6

(48)

The remaining thing is to find a rule that dominates thecoefficients of each term of the numerator of the left-handsides of (41) to (46)

Differentiating each side of (28) with respect to thevariable 119888 successively and using some suitable identitiesin Section 1 and Lemma 1 we obtain a set of infinite

Abstract and Applied Analysis 7

series involving binomial coefficients harmonic numbersand generalized harmonic numbers different from those inTheorem 2 as in the following theorem

Theorem 8 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119888 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

minus (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

3

+ 3119867(1)

119899(119888 minus 1)

times119867(2)

119899(119888 minus 1) + 2119867

(3)

119899(119888 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))3

minus 3 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887)

minus 120595(2)

(119888) minus 120595(2)

(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

4

+ 6119867(1)

119899(119888 minus 1)

2

times 119867(2)

119899(119888 minus 1) + 8119867

(1)

119899(119888 minus 1)119867

(3)

119899(119888 minus 1)

+ 3119867(2)

119899(119888 minus 1)

2

+ 6119867(4)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))4

minus 6(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887) minus 120595(2)

(119888)

minus120595(2)

(119888 minus 119886 minus 119887))

+ 3 (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888)

minus1205951015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) + 120595(3)

(119888 minus 119887) minus 120595(3)

(119888)

minus120595(3)

(119888 minus 119886 minus 119887)) ]

(49)

Setting 119888 = 1 and 119886 = 119887 = minus12 in (49) and usingsome suitable identities in Section 1 and special values of 120595-function (see eg [4 Section 12] and [5 Section 13]) weobtain a set of interesting infinite series involving binomialcoefficients and harmonic numbers given in the followingcorollary

Corollary 9 Each of the following identities holds true

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

119867119899=

4

120587(3 minus 4 log 2)

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)2+ 119867(2)

119899]

=4

120587[(3 minus 4 log 2)2 + 7 minus 4120577 (2)]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)3+ 3119867119899119867(2)

119899+ 2119867(3)

119899]

=4

120587[(3 minus 4 log 2)3 minus 3 (3 minus 4 log 2) (4120577 (2) minus 7)

minus24120577 (3) + 30]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)4+ 6(119867

119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

+ 6119867(4)

119899]

=4

120587[(3 minus 4 log 2)4 minus 6(3 minus 4 log 2)2 (4120577 (2) minus 7)

+ 24 (3 minus 4 log 2) (5 minus 4120577 (3)) + 3(4120577 (2) minus 7)2

+186 minus 168120577 (4) ]

(50)

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

1198892

1198891205722

1

(120572)119899

=1

(120572)119899

[119867(1)

119899(120572 minus 1)

2

+ 119867(2)

119899(120572 minus 1)]

(119899 isin N0 120572 isin C Z

minus

0)

119889ℓ

119889119911ℓ119867(119904)

119899(119911) = (minus1)

ℓ(119904)ℓ119867(119904+ℓ)

119899(119911)

(119899 isin N ℓ isin N0 119904 isin C Z

minus)

119889

119889119886Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

119889

119889119888Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot (120595 (119888) + 120595 (119888 minus 119886 minus 119887)

minus120595 (119888 minus 119886) minus 120595 (119888 minus 119887))

(30)

Now we are ready to present certain general identitiesof infinite series involving binomial coefficients harmonicnumbers and generalized harmonic numbers as in thefollowing theorem

Theorem 2 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

(31)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [(120595(119888 minus 119886) minus 120595(119888 minus 119886 minus 119887))

2

minus (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))]

(32)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(2)

119899(119886 minus 1)

+ 2119867(3)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

3

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887)) ]

(33)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

119867(2)

119899(119886 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(3)

119899(119886 minus 1)

+3119867(2)

119899(119886 minus 1)

2

minus 6119867(4)

119899(119886 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

4

minus 6(120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))2

times (1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119886) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 3(1205951015840(119888 minus 119886) minus 120595

1015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) minus 120595(3)

(119888 minus 119886 minus 119887)) ]

(34)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)sdot [ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

+1205951015840(119888 minus 119886 minus 119887)]

(35)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

minus119867(1)

119899(119886 minus 1)119867

(2)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

+ 2 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(2)(119888 minus 119886 minus 119887)

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))]

(36)infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(2)

119899(119887 minus 1)

+2119867(1)

119899(119886 minus 1)119867

(3)

119899(119887 minus 1) ]

Abstract and Applied Analysis 5

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

+ 3(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 1205951015840(119888 minus 119886 minus 119887) + 120595

(3)(119888 minus 119886 minus 119887)

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 3 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) minus 31205951015840(119888 minus 119886 minus 119887)

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))]

(37)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

4

minus 6119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

119867(2)

119899(119887 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(3)

119899(119887 minus 1)

+ 3119867(1)

119899(119886 minus 1) 119867

(2)

119899(119887 minus 1)

2

minus 6119867(1)

119899(119886 minus 1)119867

(4)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))4

+ 4(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(4)(119888 minus 119886 minus 119887)

minus 6 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 6(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 120595(2)

(119888 minus 119886 minus 119887)

minus 12 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(3)

(119888 minus 119886 minus 119887)

+ 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

2

+ 6 (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) + 4 1205951015840(119888 minus 119886 minus 119887)

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(3)

(119888 minus 119887) minus 120595(3)

(119888 minus 119886 minus 119887))]

(38)

Proof Differentiating each side of (28) with respect to thevariable 119886 and using some suitable identities in Section 1 andLemma 1 we obtain (31) Differentiating each side of (31) withrespect to the variable 119886 we get (32) Similarly we prove (33)and (34) Differentiating each side of (31) with respect to thevariable 119887 and using some suitable identities in Section 1 andLemma 1 we obtain (35) Similarly we prove (36) (37) and(38)

Setting 119888 = 2 and 119886 = 119887 = 1 in (31) to (34) and us-ing some suitable identities in Section 1 we obtain someinteresting identities involving harmonic numbers and gen-eralized harmonic numbers given in the following corollary

Corollary 3 Each of the following identities holds trueinfin

sum

119899=1

119867119899

(119899 + 1) (119899 + 2)= 1

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)2minus 119867(2)

119899] = 2 = 1 sdot 2

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)3minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

= 6 = 2 sdot 3

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)4

minus 6(119867119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

minus 6119867(4)

119899]

= 24 = 6 sdot 4

(39)

6 Abstract and Applied Analysis

Differentiating only the left-hand side of (34) with respectto the variable 119886 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (39) we may guess twoidentities asserted by the following conjecture

Conjecture 4 Each of the following identities may hold trueinfin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)5minus 10(119867

119899)3119867(2)

119899+ 20(119867

119899)2119867(3)

119899

+ 15119867119899(119867(2)

119899)2

minus 30119867119899119867(4)

119899

minus20119867(2)

119899119867(3)

119899+ 24119867

(5)

119899]

= 120 = 24 sdot 5

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6

minus 15(119867119899)4

119867(2)

119899+ 40(119867

119899)3

119867(3)

119899

+ 45(119867119899)2

(119867(2)

119899)2

minus 90(119867119899)2119867(4)

119899

minus 120119867119899119867(2)

119899119867(3)

119899+ 144119867

119899119867(5)

119899

minus 15(119867(2)

119899)3

+ 90119867(2)

119899119867(4)

119899+ 40(119867

(3)

119899)2

minus120119867(5)

119899] = 720 = 120 sdot 6

(40)

Setting 119888 = 3 and 119886 = 119887 = 1 in (35) to (44) and using somesuitable identities in Section 1 and Lemma 1 we obtain a setof very interesting identities in the following corollary

Corollary 5 Each of the following identities holds true

infin

sum

119899=1

(119867119899)2

(119899 + 1) (119899 + 2)= 2 (1 + 120577 (2)) (41)

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3)) (42)

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(43)

infin

sum

119899=1

(((119867119899)5minus 6(119867

119899)3119867(2)

119899+ 8(119867

119899)2

119867(3)

119899

+3119867119899(119867(2)

119899)2

minus 6119867119899119867(4)

119899)

times((119899 + 1) (119899 + 2))minus1)

= 48 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5))

(44)

Differentiating only the left-hand side of (38) with respectto the variable 119887 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (41) to (44) wemay guesstwo identities asserted by the following conjecture

Conjecture 6 Each of the following identities may hold true

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6minus 10(119867

119899)4119867(2)

119899+ 20(119867

119899)3119867(3)

119899

+ 15(119867119899)2(119867(2)

119899)2

minus 30(119867119899)2119867(4)

119899

minus20119867119899119867(2)

119899119867(3)

119899+ 24119867

119899119867(5)

119899]

= 240 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) + 120577 (6))

(45)infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)7minus 15(119867

119899)5119867(2)

119899+ 40(119867

119899)4119867(3)

119899

+ 45(119867119899)3(119867119899)2minus 90(119867

119899)3119867(4)

119899

minus 120(119867119899)2

119867(2)

119899119867(3)

119899+ 144(119867

119899)2

119867(5)

119899

minus 15119867119899(119867(2)

119899)3

+ 90119867119899119867(2)

119899119867(4)

119899

+40119867119899(119867(3)

119899)2

minus 120119867119899119867(6)

119899]

= 1440 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) +120577 (6) + 120577 (7))

(46)

Remark 7 It is interesting to observe that the number ofterms in the numerator of each of the left-hand sides of(41) to (46) is equal to the number of partitions of 119896 minus

1 (119896 = 2 3 4 5 6 7) respectively For example for (46) allthe partitions of 7 minus 1 = 6 are as follows

7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

= 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 2 + 2

= 1 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 3

= 1 + 1 + 2 + 3 = 1 + 3 + 3

= 1 + 1 + 1 + 4 = 1 + 1 + 5 = 1 + 2 + 4

(47)

where 119896 denotes119867(119896)119899

(119896 = 1 2 3 4 5) and + is translated intoa multiplication of its corresponding 119867

(119896)

119899 The coefficient of

each of the right-hand sides of (41) to (46) has the followingrule

2 4 = 2 sdot 2 12 = 4 sdot 3 48 = 12 sdot 4

240 = 48 sdot 5 1440 = 240 sdot 6

(48)

The remaining thing is to find a rule that dominates thecoefficients of each term of the numerator of the left-handsides of (41) to (46)

Differentiating each side of (28) with respect to thevariable 119888 successively and using some suitable identitiesin Section 1 and Lemma 1 we obtain a set of infinite

Abstract and Applied Analysis 7

series involving binomial coefficients harmonic numbersand generalized harmonic numbers different from those inTheorem 2 as in the following theorem

Theorem 8 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119888 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

minus (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

3

+ 3119867(1)

119899(119888 minus 1)

times119867(2)

119899(119888 minus 1) + 2119867

(3)

119899(119888 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))3

minus 3 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887)

minus 120595(2)

(119888) minus 120595(2)

(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

4

+ 6119867(1)

119899(119888 minus 1)

2

times 119867(2)

119899(119888 minus 1) + 8119867

(1)

119899(119888 minus 1)119867

(3)

119899(119888 minus 1)

+ 3119867(2)

119899(119888 minus 1)

2

+ 6119867(4)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))4

minus 6(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887) minus 120595(2)

(119888)

minus120595(2)

(119888 minus 119886 minus 119887))

+ 3 (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888)

minus1205951015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) + 120595(3)

(119888 minus 119887) minus 120595(3)

(119888)

minus120595(3)

(119888 minus 119886 minus 119887)) ]

(49)

Setting 119888 = 1 and 119886 = 119887 = minus12 in (49) and usingsome suitable identities in Section 1 and special values of 120595-function (see eg [4 Section 12] and [5 Section 13]) weobtain a set of interesting infinite series involving binomialcoefficients and harmonic numbers given in the followingcorollary

Corollary 9 Each of the following identities holds true

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

119867119899=

4

120587(3 minus 4 log 2)

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)2+ 119867(2)

119899]

=4

120587[(3 minus 4 log 2)2 + 7 minus 4120577 (2)]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)3+ 3119867119899119867(2)

119899+ 2119867(3)

119899]

=4

120587[(3 minus 4 log 2)3 minus 3 (3 minus 4 log 2) (4120577 (2) minus 7)

minus24120577 (3) + 30]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)4+ 6(119867

119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

+ 6119867(4)

119899]

=4

120587[(3 minus 4 log 2)4 minus 6(3 minus 4 log 2)2 (4120577 (2) minus 7)

+ 24 (3 minus 4 log 2) (5 minus 4120577 (3)) + 3(4120577 (2) minus 7)2

+186 minus 168120577 (4) ]

(50)

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

+ 3(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 1205951015840(119888 minus 119886 minus 119887) + 120595

(3)(119888 minus 119886 minus 119887)

minus 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 3 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) minus 31205951015840(119888 minus 119886 minus 119887)

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))]

(37)

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

4

minus 6119867(1)

119899(119886 minus 1) 119867

(1)

119899(119887 minus 1)

2

119867(2)

119899(119887 minus 1)

+ 8119867(1)

119899(119886 minus 1)119867

(1)

119899(119887 minus 1)119867

(3)

119899(119887 minus 1)

+ 3119867(1)

119899(119886 minus 1) 119867

(2)

119899(119887 minus 1)

2

minus 6119867(1)

119899(119886 minus 1)119867

(4)

119899(119887 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)[ (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))4

+ 4(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))3

times 1205951015840(119888 minus 119886 minus 119887) minus 120595

(4)(119888 minus 119886 minus 119887)

minus 6 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

minus 6(120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))2

times 120595(2)

(119888 minus 119886 minus 119887)

minus 12 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119887) minus 120595 (119888 minus 119886 minus 119887))

times 120595(3)

(119888 minus 119886 minus 119887)

+ 3 (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

2

+ 6 (1205951015840(119888 minus 119887) minus 120595

1015840(119888 minus 119886 minus 119887))

times 120595(2)

(119888 minus 119886 minus 119887) + 4 1205951015840(119888 minus 119886 minus 119887)

times (120595(2)

(119888 minus 119887) minus 120595(2)

(119888 minus 119886 minus 119887))

minus (120595 (119888 minus 119886) minus 120595 (119888 minus 119886 minus 119887))

times (120595(3)

(119888 minus 119887) minus 120595(3)

(119888 minus 119886 minus 119887))]

(38)

Proof Differentiating each side of (28) with respect to thevariable 119886 and using some suitable identities in Section 1 andLemma 1 we obtain (31) Differentiating each side of (31) withrespect to the variable 119886 we get (32) Similarly we prove (33)and (34) Differentiating each side of (31) with respect to thevariable 119887 and using some suitable identities in Section 1 andLemma 1 we obtain (35) Similarly we prove (36) (37) and(38)

Setting 119888 = 2 and 119886 = 119887 = 1 in (31) to (34) and us-ing some suitable identities in Section 1 we obtain someinteresting identities involving harmonic numbers and gen-eralized harmonic numbers given in the following corollary

Corollary 3 Each of the following identities holds trueinfin

sum

119899=1

119867119899

(119899 + 1) (119899 + 2)= 1

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)2minus 119867(2)

119899] = 2 = 1 sdot 2

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)3minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

= 6 = 2 sdot 3

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)4

minus 6(119867119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

minus 6119867(4)

119899]

= 24 = 6 sdot 4

(39)

6 Abstract and Applied Analysis

Differentiating only the left-hand side of (34) with respectto the variable 119886 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (39) we may guess twoidentities asserted by the following conjecture

Conjecture 4 Each of the following identities may hold trueinfin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)5minus 10(119867

119899)3119867(2)

119899+ 20(119867

119899)2119867(3)

119899

+ 15119867119899(119867(2)

119899)2

minus 30119867119899119867(4)

119899

minus20119867(2)

119899119867(3)

119899+ 24119867

(5)

119899]

= 120 = 24 sdot 5

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6

minus 15(119867119899)4

119867(2)

119899+ 40(119867

119899)3

119867(3)

119899

+ 45(119867119899)2

(119867(2)

119899)2

minus 90(119867119899)2119867(4)

119899

minus 120119867119899119867(2)

119899119867(3)

119899+ 144119867

119899119867(5)

119899

minus 15(119867(2)

119899)3

+ 90119867(2)

119899119867(4)

119899+ 40(119867

(3)

119899)2

minus120119867(5)

119899] = 720 = 120 sdot 6

(40)

Setting 119888 = 3 and 119886 = 119887 = 1 in (35) to (44) and using somesuitable identities in Section 1 and Lemma 1 we obtain a setof very interesting identities in the following corollary

Corollary 5 Each of the following identities holds true

infin

sum

119899=1

(119867119899)2

(119899 + 1) (119899 + 2)= 2 (1 + 120577 (2)) (41)

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3)) (42)

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(43)

infin

sum

119899=1

(((119867119899)5minus 6(119867

119899)3119867(2)

119899+ 8(119867

119899)2

119867(3)

119899

+3119867119899(119867(2)

119899)2

minus 6119867119899119867(4)

119899)

times((119899 + 1) (119899 + 2))minus1)

= 48 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5))

(44)

Differentiating only the left-hand side of (38) with respectto the variable 119887 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (41) to (44) wemay guesstwo identities asserted by the following conjecture

Conjecture 6 Each of the following identities may hold true

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6minus 10(119867

119899)4119867(2)

119899+ 20(119867

119899)3119867(3)

119899

+ 15(119867119899)2(119867(2)

119899)2

minus 30(119867119899)2119867(4)

119899

minus20119867119899119867(2)

119899119867(3)

119899+ 24119867

119899119867(5)

119899]

= 240 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) + 120577 (6))

(45)infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)7minus 15(119867

119899)5119867(2)

119899+ 40(119867

119899)4119867(3)

119899

+ 45(119867119899)3(119867119899)2minus 90(119867

119899)3119867(4)

119899

minus 120(119867119899)2

119867(2)

119899119867(3)

119899+ 144(119867

119899)2

119867(5)

119899

minus 15119867119899(119867(2)

119899)3

+ 90119867119899119867(2)

119899119867(4)

119899

+40119867119899(119867(3)

119899)2

minus 120119867119899119867(6)

119899]

= 1440 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) +120577 (6) + 120577 (7))

(46)

Remark 7 It is interesting to observe that the number ofterms in the numerator of each of the left-hand sides of(41) to (46) is equal to the number of partitions of 119896 minus

1 (119896 = 2 3 4 5 6 7) respectively For example for (46) allthe partitions of 7 minus 1 = 6 are as follows

7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

= 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 2 + 2

= 1 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 3

= 1 + 1 + 2 + 3 = 1 + 3 + 3

= 1 + 1 + 1 + 4 = 1 + 1 + 5 = 1 + 2 + 4

(47)

where 119896 denotes119867(119896)119899

(119896 = 1 2 3 4 5) and + is translated intoa multiplication of its corresponding 119867

(119896)

119899 The coefficient of

each of the right-hand sides of (41) to (46) has the followingrule

2 4 = 2 sdot 2 12 = 4 sdot 3 48 = 12 sdot 4

240 = 48 sdot 5 1440 = 240 sdot 6

(48)

The remaining thing is to find a rule that dominates thecoefficients of each term of the numerator of the left-handsides of (41) to (46)

Differentiating each side of (28) with respect to thevariable 119888 successively and using some suitable identitiesin Section 1 and Lemma 1 we obtain a set of infinite

Abstract and Applied Analysis 7

series involving binomial coefficients harmonic numbersand generalized harmonic numbers different from those inTheorem 2 as in the following theorem

Theorem 8 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119888 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

minus (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

3

+ 3119867(1)

119899(119888 minus 1)

times119867(2)

119899(119888 minus 1) + 2119867

(3)

119899(119888 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))3

minus 3 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887)

minus 120595(2)

(119888) minus 120595(2)

(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

4

+ 6119867(1)

119899(119888 minus 1)

2

times 119867(2)

119899(119888 minus 1) + 8119867

(1)

119899(119888 minus 1)119867

(3)

119899(119888 minus 1)

+ 3119867(2)

119899(119888 minus 1)

2

+ 6119867(4)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))4

minus 6(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887) minus 120595(2)

(119888)

minus120595(2)

(119888 minus 119886 minus 119887))

+ 3 (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888)

minus1205951015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) + 120595(3)

(119888 minus 119887) minus 120595(3)

(119888)

minus120595(3)

(119888 minus 119886 minus 119887)) ]

(49)

Setting 119888 = 1 and 119886 = 119887 = minus12 in (49) and usingsome suitable identities in Section 1 and special values of 120595-function (see eg [4 Section 12] and [5 Section 13]) weobtain a set of interesting infinite series involving binomialcoefficients and harmonic numbers given in the followingcorollary

Corollary 9 Each of the following identities holds true

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

119867119899=

4

120587(3 minus 4 log 2)

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)2+ 119867(2)

119899]

=4

120587[(3 minus 4 log 2)2 + 7 minus 4120577 (2)]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)3+ 3119867119899119867(2)

119899+ 2119867(3)

119899]

=4

120587[(3 minus 4 log 2)3 minus 3 (3 minus 4 log 2) (4120577 (2) minus 7)

minus24120577 (3) + 30]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)4+ 6(119867

119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

+ 6119867(4)

119899]

=4

120587[(3 minus 4 log 2)4 minus 6(3 minus 4 log 2)2 (4120577 (2) minus 7)

+ 24 (3 minus 4 log 2) (5 minus 4120577 (3)) + 3(4120577 (2) minus 7)2

+186 minus 168120577 (4) ]

(50)

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Differentiating only the left-hand side of (34) with respectto the variable 119886 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (39) we may guess twoidentities asserted by the following conjecture

Conjecture 4 Each of the following identities may hold trueinfin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)5minus 10(119867

119899)3119867(2)

119899+ 20(119867

119899)2119867(3)

119899

+ 15119867119899(119867(2)

119899)2

minus 30119867119899119867(4)

119899

minus20119867(2)

119899119867(3)

119899+ 24119867

(5)

119899]

= 120 = 24 sdot 5

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6

minus 15(119867119899)4

119867(2)

119899+ 40(119867

119899)3

119867(3)

119899

+ 45(119867119899)2

(119867(2)

119899)2

minus 90(119867119899)2119867(4)

119899

minus 120119867119899119867(2)

119899119867(3)

119899+ 144119867

119899119867(5)

119899

minus 15(119867(2)

119899)3

+ 90119867(2)

119899119867(4)

119899+ 40(119867

(3)

119899)2

minus120119867(5)

119899] = 720 = 120 sdot 6

(40)

Setting 119888 = 3 and 119886 = 119887 = 1 in (35) to (44) and using somesuitable identities in Section 1 and Lemma 1 we obtain a setof very interesting identities in the following corollary

Corollary 5 Each of the following identities holds true

infin

sum

119899=1

(119867119899)2

(119899 + 1) (119899 + 2)= 2 (1 + 120577 (2)) (41)

infin

sum

119899=1

(119867119899)3minus 119867119899119867(2)

119899

(119899 + 1) (119899 + 2)= 4 (1 + 120577 (2) + 120577 (3)) (42)

infin

sum

119899=1

(119867119899)4minus 3(119867

119899)2119867(2)

119899+ 2119867119899119867(3)

119899

(119899 + 1) (119899 + 2)

= 12 (1 + 120577 (2) + 120577 (3) + 120577 (4))

(43)

infin

sum

119899=1

(((119867119899)5minus 6(119867

119899)3119867(2)

119899+ 8(119867

119899)2

119867(3)

119899

+3119867119899(119867(2)

119899)2

minus 6119867119899119867(4)

119899)

times((119899 + 1) (119899 + 2))minus1)

= 48 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5))

(44)

Differentiating only the left-hand side of (38) with respectto the variable 119887 twice and setting 119888 = 3 and 119886 = 119887 = 1 in eachexpression in view of the identities (41) to (44) wemay guesstwo identities asserted by the following conjecture

Conjecture 6 Each of the following identities may hold true

infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)6minus 10(119867

119899)4119867(2)

119899+ 20(119867

119899)3119867(3)

119899

+ 15(119867119899)2(119867(2)

119899)2

minus 30(119867119899)2119867(4)

119899

minus20119867119899119867(2)

119899119867(3)

119899+ 24119867

119899119867(5)

119899]

= 240 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) + 120577 (6))

(45)infin

sum

119899=1

1

(119899 + 1) (119899 + 2)[(119867119899)7minus 15(119867

119899)5119867(2)

119899+ 40(119867

119899)4119867(3)

119899

+ 45(119867119899)3(119867119899)2minus 90(119867

119899)3119867(4)

119899

minus 120(119867119899)2

119867(2)

119899119867(3)

119899+ 144(119867

119899)2

119867(5)

119899

minus 15119867119899(119867(2)

119899)3

+ 90119867119899119867(2)

119899119867(4)

119899

+40119867119899(119867(3)

119899)2

minus 120119867119899119867(6)

119899]

= 1440 (1 + 120577 (2) + 120577 (3) + 120577 (4) + 120577 (5) +120577 (6) + 120577 (7))

(46)

Remark 7 It is interesting to observe that the number ofterms in the numerator of each of the left-hand sides of(41) to (46) is equal to the number of partitions of 119896 minus

1 (119896 = 2 3 4 5 6 7) respectively For example for (46) allthe partitions of 7 minus 1 = 6 are as follows

7 = 1 + 1 + 1 + 1 + 1 + 1 + 1

= 1 + 1 + 1 + 1 + 1 + 2 = 1 + 1 + 1 + 2 + 2

= 1 + 2 + 2 + 2 = 1 + 1 + 1 + 1 + 3

= 1 + 1 + 2 + 3 = 1 + 3 + 3

= 1 + 1 + 1 + 4 = 1 + 1 + 5 = 1 + 2 + 4

(47)

where 119896 denotes119867(119896)119899

(119896 = 1 2 3 4 5) and + is translated intoa multiplication of its corresponding 119867

(119896)

119899 The coefficient of

each of the right-hand sides of (41) to (46) has the followingrule

2 4 = 2 sdot 2 12 = 4 sdot 3 48 = 12 sdot 4

240 = 48 sdot 5 1440 = 240 sdot 6

(48)

The remaining thing is to find a rule that dominates thecoefficients of each term of the numerator of the left-handsides of (41) to (46)

Differentiating each side of (28) with respect to thevariable 119888 successively and using some suitable identitiesin Section 1 and Lemma 1 we obtain a set of infinite

Abstract and Applied Analysis 7

series involving binomial coefficients harmonic numbersand generalized harmonic numbers different from those inTheorem 2 as in the following theorem

Theorem 8 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119888 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

minus (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

3

+ 3119867(1)

119899(119888 minus 1)

times119867(2)

119899(119888 minus 1) + 2119867

(3)

119899(119888 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))3

minus 3 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887)

minus 120595(2)

(119888) minus 120595(2)

(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

4

+ 6119867(1)

119899(119888 minus 1)

2

times 119867(2)

119899(119888 minus 1) + 8119867

(1)

119899(119888 minus 1)119867

(3)

119899(119888 minus 1)

+ 3119867(2)

119899(119888 minus 1)

2

+ 6119867(4)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))4

minus 6(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887) minus 120595(2)

(119888)

minus120595(2)

(119888 minus 119886 minus 119887))

+ 3 (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888)

minus1205951015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) + 120595(3)

(119888 minus 119887) minus 120595(3)

(119888)

minus120595(3)

(119888 minus 119886 minus 119887)) ]

(49)

Setting 119888 = 1 and 119886 = 119887 = minus12 in (49) and usingsome suitable identities in Section 1 and special values of 120595-function (see eg [4 Section 12] and [5 Section 13]) weobtain a set of interesting infinite series involving binomialcoefficients and harmonic numbers given in the followingcorollary

Corollary 9 Each of the following identities holds true

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

119867119899=

4

120587(3 minus 4 log 2)

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)2+ 119867(2)

119899]

=4

120587[(3 minus 4 log 2)2 + 7 minus 4120577 (2)]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)3+ 3119867119899119867(2)

119899+ 2119867(3)

119899]

=4

120587[(3 minus 4 log 2)3 minus 3 (3 minus 4 log 2) (4120577 (2) minus 7)

minus24120577 (3) + 30]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)4+ 6(119867

119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

+ 6119867(4)

119899]

=4

120587[(3 minus 4 log 2)4 minus 6(3 minus 4 log 2)2 (4120577 (2) minus 7)

+ 24 (3 minus 4 log 2) (5 minus 4120577 (3)) + 3(4120577 (2) minus 7)2

+186 minus 168120577 (4) ]

(50)

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

series involving binomial coefficients harmonic numbersand generalized harmonic numbers different from those inTheorem 2 as in the following theorem

Theorem 8 Each of the following summation formulas holdstrue

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

119867(1)

119899(119888 minus 1)

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

minus (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

3

+ 3119867(1)

119899(119888 minus 1)

times119867(2)

119899(119888 minus 1) + 2119867

(3)

119899(119888 minus 1) ]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))3

minus 3 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887)

minus 120595(2)

(119888) minus 120595(2)

(119888 minus 119886 minus 119887))]

infin

sum

119899=1

(119886)119899(119887)119899

119899(119888)119899

[119867(1)

119899(119888 minus 1)

4

+ 6119867(1)

119899(119888 minus 1)

2

times 119867(2)

119899(119888 minus 1) + 8119867

(1)

119899(119888 minus 1)119867

(3)

119899(119888 minus 1)

+ 3119867(2)

119899(119888 minus 1)

2

+ 6119867(4)

119899(119888 minus 1)]

=Γ (119888) Γ (119888 minus 119886 minus 119887)

Γ (119888 minus 119886) Γ (119888 minus 119887)

times [(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))4

minus 6(120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))2

sdot (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888) minus 120595

1015840(119888 minus 119886 minus 119887))

+ 4 (120595 (119888 minus 119886) + 120595 (119888 minus 119887) minus 120595 (119888) minus 120595 (119888 minus 119886 minus 119887))

sdot (120595(2)

(119888 minus 119886) + 120595(2)

(119888 minus 119887) minus 120595(2)

(119888)

minus120595(2)

(119888 minus 119886 minus 119887))

+ 3 (1205951015840(119888 minus 119886) + 120595

1015840(119888 minus 119887) minus 120595

1015840(119888)

minus1205951015840(119888 minus 119886 minus 119887))

2

minus (120595(3)

(119888 minus 119886) + 120595(3)

(119888 minus 119887) minus 120595(3)

(119888)

minus120595(3)

(119888 minus 119886 minus 119887)) ]

(49)

Setting 119888 = 1 and 119886 = 119887 = minus12 in (49) and usingsome suitable identities in Section 1 and special values of 120595-function (see eg [4 Section 12] and [5 Section 13]) weobtain a set of interesting infinite series involving binomialcoefficients and harmonic numbers given in the followingcorollary

Corollary 9 Each of the following identities holds true

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

119867119899=

4

120587(3 minus 4 log 2)

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)2+ 119867(2)

119899]

=4

120587[(3 minus 4 log 2)2 + 7 minus 4120577 (2)]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)3+ 3119867119899119867(2)

119899+ 2119867(3)

119899]

=4

120587[(3 minus 4 log 2)3 minus 3 (3 minus 4 log 2) (4120577 (2) minus 7)

minus24120577 (3) + 30]

infin

sum

119899=1

(2119899

119899)2

(2119899 minus 1)224119899

[(119867119899)4+ 6(119867

119899)2119867(2)

119899

+ 8119867119899119867(3)

119899+ 3(119867

(2)

119899)2

+ 6119867(4)

119899]

=4

120587[(3 minus 4 log 2)4 minus 6(3 minus 4 log 2)2 (4120577 (2) minus 7)

+ 24 (3 minus 4 log 2) (5 minus 4120577 (3)) + 3(4120577 (2) minus 7)2

+186 minus 168120577 (4) ]

(50)

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

3 Finite Series Involving BinomialCoefficients Harmonic Numbers andGeneralized Harmonic Numbers

Setting 119887 = minus119873 isin N in some chosen formulas in Theorems 2and 8 and using some suitable identities in Section 1 and thefollowing known and easily derivable formula

(minus119873)119899=

(minus1)119899119873

(119873 minus 119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

=

(minus1)119899119899 (

119873

119899) (0 le 119899 le 119873 119873 isin N)

0 (119899 gt 119873)

(51)

we obtain a set of finite series involving binomial coefficientsharmonic numbers and generalized harmonic numbersgiven in the following theorem

Theorem 10 Each of the following finite summation formulasholds true

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119886 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)119867(1)

119873(119888 minus 119886 minus 1) (119873 isin N

0)

(52)

where the empty sum is (as usual) understood to be nilthroughout this paper

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

2

minus 119867(2)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

2

minus119867(2)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

3

minus 3119867(1)

119899(119886 minus 1)

times 119867(2)

119899(119886 minus 1) + 2119867

(3)

119899(119886 minus 1) ]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

3

minus 3119867(1)

119873(119888 minus 119886 minus 1)

times 119867(2)

119873(119888 minus 119886 minus 1)

+2119867(3)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899) [119867

(1)

119899(119886 minus 1)

4

minus 6119867(1)

119899(119886 minus 1)

2

times 119867(2)

119899(119886 minus 1) + 8119867

(1)

119899(119886 minus 1)

times 119867(3)

119899(119886 minus 1) + 3119867

(2)

119899(119886 minus 1)

2

minus6119867(4)

119899(119886 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 119886 minus 1)

4

minus 6119867(1)

119873(119888 minus 119886 minus 1)

2

times 119867(2)

119873(119888 minus 119886 minus 1)

+ 8119867(1)

119873(119888 minus 119886 minus 1)119867

(3)

119873(119888 minus 119886 minus 1)

+ 3119867(2)

119873(119888 minus 119886 minus 1)

2

minus6119867(4)

119873(119888 minus 119886 minus 1) ] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899 (119886)119899

(119888)119899

(119873

119899)119867(1)

119899(119888 minus 1)

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)]

(119873 isin N0)

119873

sum

119899=1

(minus1)119899(119873

119899)(119886)119899

(119888)119899

[119867(1)

119899(119888 minus 1)

2

+ 119867(2)

119899(119888 minus 1)]

=Γ (119888 minus 119886 + 119873) Γ (119888)

Γ (119888 minus 119886) Γ (119888 + 119873)[119867(1)

119873(119888 minus 1) minus 119867

(1)

119873(119888 minus 119886 minus 1)

2

+119867(2)

119873(119888 minus 1) minus 119867

(2)

119873(119888 minus 119886 minus 1) ]

(119873 isin N0)

(53)

Setting 119886 = 1 and 119888 = 2 in (52) to (53) and usingsome suitable identities in Section 1 we obtain a set of inter-esting identities involving binomial coefficients harmonicnumbers and generalized harmonic numbers given in thefollowing corollary

Corollary 11 Each of the following identities holds true

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867119899=

119867119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)2minus 119867(2)

119899] =

(119867119873)2minus 119867(2)

119873

119873 + 1(119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [(119867

119899)3

minus 3119867119899119867(2)

119899+ 2119867(3)

119899]

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

=1

119873 + 1[(119867119873)3minus 3119867119873119867(2)

119873+ 2119867(3)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899

119899 + 1(119873

119899) [(119867

119899)4

minus 6(119867119899)2

119867(2)

119899+ 8119867119899119867(3)

119899

+3(119867(2)

119899)2

minus 6119867(4)

119899]

=1

119873 + 1[(119867119873)4minus 6(119867

119873)2119867(2)

119873+ 8119867119873119867(3)

119873

+3(119867(2)

119873)2

minus 6119867(4)

119873] (119873 isin N

0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899)119867(1)

119899(1) =

119873

(119873 + 1)2

(119873 isin N0)

119873

sum

119899=1

(minus1)119899+1

119899 + 1(119873

119899) [119867

(1)

119899(1)2

+ 119867(2)

119899(1)]

=2119873

(119873 + 1)3

(119873 isin N0)

(54)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by Dongguk University ResearchFund of 2013 Also it was supported in part by BasicScience Research Program through the National ResearchFoundation of Korea funded by the Ministry of EducationScience and Technology (2010-0011005)

References

[1] V S Adamchik and H M Srivastava ldquoSome series of the zetaand related functionsrdquo Analysis vol 18 no 2 pp 131ndash144 1998

[2] J Choi and H M Srivastava ldquoSome summation formulasinvolving harmonic numbers and generalized harmonic num-bersrdquo Mathematical and Computer Modelling vol 54 no 9-10pp 2220ndash2234 2011

[3] R L Graham D E Knuth and O Patashnik Concrete Math-ematics A Foundation for Computer Science Addison-WesleyReading Mass USA 2nd edition 1989

[4] HM Srivastava and J Choi Series Associated with the Zeta andRelated Functions Kluwer Academic London UK 2001

[5] H M Srivastava and J Choi Zeta and q-Zeta Functions andAssociated Series and Integrals Elsevier Science AmsterdamThe Netherlands 2012

[6] J Choi ldquoCertain summation formulas involving harmonicnumbers and generalized harmonic numbersrdquo Applied Math-ematics and Computation vol 218 no 3 pp 734ndash740 2011

[7] T M Rassias and H M Srivastava ldquoSome classes of infiniteseries associated with the Riemann Zeta and Polygamma func-tions and generalized harmonic numbersrdquoAppliedMathematicsand Computation vol 131 no 2-3 pp 593ndash605 2002

[8] A Sofo and H M Srivastava ldquoIdentities for the harmonicnumbers and binomial coefficientsrdquo Ramanujan Journal vol25 no 1 pp 93ndash113 2011

[9] M W Coffey ldquoOn some series representations of the Hurwitzzeta functionrdquo Journal of Computational and Applied Mathe-matics vol 216 no 1 pp 297ndash305 2008

[10] B C Berndt Ramanujanrsquos Notebooks Part I Springer NewYork NY USA 1985

[11] D Borwein and J M Borwein ldquoOn an intriguing integraland some series related to 120577(4)rdquo Proceedings of the AmericanMathematical Society vol 123 no 4 pp 1191ndash1198 1995

[12] N Nielsen Die Gammafunktion Chelsea New York NY USA1965

[13] P Flajolet and B Salvy ldquoEuler sums and contour integralrepresentationsrdquo ExperimentalMathematics vol 7 no 1 pp 15ndash35 1998

[14] D H Bailey J M Borwein and R Girgensohn ldquoExperimentalevaluation of Euler sumsrdquo ExperimentalMathematics vol 3 no1 pp 17ndash30 1994

[15] P J de Doelder ldquoOn some series containing 120595(119909) minus 120595(119910 gt)

and (120595 (119909) minus 120595 (119910 gt))2 for certain values of 119909 and 119910rdquo Journal

of Computational and Applied Mathematics vol 37 no 1ndash3 pp125ndash141 1991

[16] M E Hoffman ldquoMultiple harmonic seriesrdquo Pacific Journal ofMathematics vol 152 no 2 pp 275ndash290 1992

[17] D Zagier ldquoValues of Zeta functions and their applicationsrdquo inProceedings of the 1st EuropeanCongress ofMathematics VolumeII (Paris France 1992) A Joseph F Mignot F Murat B Prumand R Rentschler Eds Progress in Mathematics 120 pp 497ndash512 Birkhaauser Basel Switzerland 1994

[18] D Bowman and D M Bradley Multiple Polylogarithms Abrief survey in q-Series with Applications to Combinatorics NumberTheory and Physics (Papers from the Conference heldat the University of Illinois Urbana Ill USA October 2000)(B C Berndt and K Ono Editors) pp 71ndash92 ContemporaryMathematics 291 AmericanMathematical Society ProvidenceRI USA 2001

[19] L Shen ldquoRemarks on some integrals and series involvingthe Stirling numbers and 120577(119899)rdquo Transactions of the AmericanMathematical Society vol 347 no 4 pp 1391ndash1399 1995

[20] J Choi and H M Srivastava ldquoCertain classes of infinite seriesrdquoMonatshefte fur Mathematik vol 127 no 1 pp 15ndash25 1999

[21] J Choi and H M Srivastava ldquoExplicit evaluation of Euler andrelated sumsrdquo Ramanujan Journal vol 10 no 1 pp 51ndash70 2005

[22] W-C Chu ldquoHypergeometric series and the Riemann zetafunctionrdquo Acta Arithmetica vol 82 no 2 pp 103ndash118 1997

[23] WChu and L deDonno ldquoHypergeometric series and harmonicnumber identitiesrdquo Advances in Applied Mathematics vol 34no 1 pp 123ndash137 2005

[24] H Alzer D Karayannakis and H M Srivastava ldquoSeriesrepresentations for some mathematical constantsrdquo Journal ofMathematical Analysis and Applications vol 320 no 1 pp 145ndash162 2006

[25] J H Conway and R K Guy The Book of Numbers SpringerBerlin Germany 1996

[26] D Cvijovic ldquoThe Dattoli-Srivastava conjectures concerninggenerating functions involving the harmonic numbersrdquoAppliedMathematics and Computation vol 215 no 11 pp 4040ndash40432010

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[27] G Dattoli and H M Srivastava ldquoA note on harmonic numbersumbral calculus and generating functionsrdquo Applied Mathemat-ics Letters vol 21 no 7 pp 686ndash693 2008

[28] D H Greene andD E Knuth Birkhauser Berlin Germany 3rdedition 1990

[29] P Paule and C Schneider ldquoComputer proofs of a new family ofharmonic number identitiesrdquoAdvances in AppliedMathematicsvol 31 no 2 pp 359ndash378 2003

[30] C Schneider ldquoSolving parameterized linear difference equa-tions inprodΣ-fieldsrdquo Technical Report 02-03 RISC-Linz Johan-nes Kepler University Linz Austria 2002 httpwwwriscuni-linzacatresearchcombinatriscpublications

[31] M Z Spivey ldquoCombinatorial sums and finite differencesrdquoDiscrete Mathematics vol 307 no 24 pp 3130ndash3146 2007

[32] WolframMathWorld httpmathworldwolframcomHarmo-nicNumberhtml

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of