Hindawi Publishing CorporationISRN Discrete MathematicsVolume 2013 Article ID 274697 7 pageshttpdxdoiorg1011552013274697

Research ArticleAn Asymptotic Formula for 119903-Bell Numbers with Real Arguments

Cristina B Corcino1 and Roberto B Corcino2

1 Institute of Mathematics University of the Philippines Diliman Quezon City 1004 Philippines2 Department of Mathematics Mindanao State University Marawi City 9700 Philippines

Correspondence should be addressed to Roberto B Corcino rcorcinoyahoocom

Received 24 December 2012 Accepted 15 January 2013

Academic Editors A Ashrafi and H Deng

Copyright copy 2013 C B Corcino and R B Corcino This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The r-Bell numbers are generalized using the concept of the Hankel contour Some properties parallel to those of the ordinary Bellnumbers are established Moreover an asymptotic approximation for 119903-Bell numbers with real arguments is obtained

1 Introduction

The 119903-Stirling numbers of the second kind denoted by 119899119896119903

are defined by Broder in [1] combinatorially to be thenumber of partitions of the set 1 2 119899 into 119896 nonemptysubsets such that the numbers 1 2 119903 are in distinctsubsets Several properties of these numbers are established in[1ndash3] Further generalization was established in [4] which iscalled (119903 120573)-Stirling numbers These numbers are equivalentto the 119903-Whitney numbers of the second kind [5] and theRucinski-Voigt numbers [6]

The sumof 119903-Stirling numbers of the second kind for inte-gral arguments was first considered by Corcino in [7] andwascalled the 119903-Bell numbers Corcino obtained an asymptoticapproximation of 119903-Bell numbers using the method of Moserand Wyman Here we use 119861

119899119903to denote the 119903-Bell numbers

that is

119861119899 119903=

119899

sum

119896=0

119899 + 119903

119896 + 119903

119903

(1)

In a followup study of Mezo [8] the 119903-Bell numbers 119861119899 119903

were given more properties One of these is the followingexponential generating function

sum

119899ge0

119861119899 119903

119905119899

119899

= exp (119903119905 + (119890119905 minus 1)) (2)

A more general form of Bell numbers denoted by 119866119909119903120573

was defined in [9] as

119866119909119903120573=

119909

2120587119894

int

H

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1 (3)

where the parameters 119909 119903 and 120573 are complex numbers withreal119903 gt 0 real120573 gt 0 and 119909 = Γ(119909 + 1) In this paper we definethe 119903-Bell numbers with complex argument using the conceptof Hankel contour and establish some properties parallelto those obtained by Mezo in [8] Moreover an asymptoticformula of these numbers for real arguments will be derivedusing the method of Moser and Wyman [10]

2 119903-Stirling Numbers of the Second Kind

Graham et al [11] proposed another way of generalizing theStirling numbers by extending the range of values of theparameters 119899 and 119896 to complex numbers This problem wasfirst considered by Flajolet and Prodinger [12] by definingthe classical Stirling numbers with complex arguments usingthe concept of Hankel contour Recently the (119903 120573)-Stirlingnumbers with complex arguments denoted by 119909119910

120573119903 were

defined in [9] by means of the following integral representa-tion over a Hankel contourH

119909

119910

120573 119903

=

119909

1199101205731199102120587119894

int

H

119890119903119911

(119890120573119911

minus 1)

119910 119889119911

119911119909+1 (4)

where 119903 and120573 are complex numberswithreal119903 gt 0real120573 gt 0 and119909 = Γ(119909 + 1) We know that for integral case the 119903-Stirling

2 ISRN Discrete Mathematics

numbers of the second kind may be obtained by taking 120573 =1 Hence using (4) we can define the second-kind 119903-Stirlingnumbers with complex arguments as follows

Definition 1 The 119903-Stirling numbers of the second kind119909+119903

119910+119903 119903of complex arguments 119909 and 119910 are defined by

119909 + 119903

119910 + 119903

119903

=

119909

1199102120587119894

int

H

119890119903119911

(119890119911

minus 1)119910 119889119911

119911119909+1 (5)

where 119903 is complex number with real119903 gt 0 and 119909 = Γ(119909 + 1)and the logarithm involved in the functions (119890119911 minus 1)119910 and 119911119909+1is taken to be the principal branch The Hankel contour Hstarts from minusinfin below the negative axis surrounds the origincounterclockwise and returns tominusinfin in the half planeimage119911 gt 0such that it has a distance 120598 le 1 from the nonpositive real axis

Remark 2 Since real119903 gt 0 and real119911 rarr minusinfin the integral in (5)converges for all values of 119909 and 119910 Also 119909+119903119910+119903

119903is a mero-

morphic function of 119909 (for any fixed 119910) with poles at thenonpositive integers and it is entire as a function of 119910 (forany fixed 119909 not a negative integer)

Remark 3 By the change of variable on the integral in (4) say119908 = 120573119911 we can express 119909119910

120573119903in terms of 119909+119903119910+119903

119903as follows

119909

119910

120573119903

= 120573119909minus119910

119909 +

119903

120573

119910 +

119903

120573

119903120573

(6)

Because of relation (6) every property of (119903 120573)-Stirlingnumbers with complex arguments will have a correspondingproperty for 119903-Stirling numbers with complex arguments andvice versa For instance the (119903 120573)-Stirling numbers in [9]satisfy the following relation

119909

119910

120573 119903

=

119909 minus 1

119910 minus 1

120573119903

+ (120573119910 + 119903)

119909 minus 1

119910

120573119903

(7)

Replacing 119903 with 120573119903 we get

119909

119910

120573 120573119903

=

119909 minus 1

119910 minus 1

120573120573119903

+ (120573119910 + 120573119903)

119909 minus 1

119910

120573120573119903

(8)

Using (6) we obtain

120573119909minus119910

119909 +

120573119903

120573

119910 +

120573119903

120573

120573119903120573

= 120573119909minus119910

119909 + (

120573119903

120573

) minus 1

119910 + (

120573119903

120573

) minus 1

120573119903120573

+ 120573119909minus1minus119910

(120573119910 + 120573119903)

119909 + (

120573119903

120573

) minus 1

119910 + 120573119903

120573119903120573

(9)

Thus we have

119909 + 119903

119910 + 119903

119903

=

119909 + 119903 minus 1

119910 + 119903 minus 1

119903

+ (119910 + 119903)

119909 + 119903 minus 1

119910 + 119903

119903

(10)

On the other hand the (119903 120573)-Stirling numbers in [9]satisfy the relation

119909

119910

120573 119903

=

1

120573119910119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (120573119895 + 119903)

119909

(11)

Again replacing 119903 with 120573119903 we obtain

119909

119910

120573120573119903

=

1

120573119910119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (120573119895 + 120573119903)

119909

(12)

By (6) we get

120573119909minus119910

119909 +

120573119903

120573

119910 +

120573119903

120573

120573119903120573

=

1

120573119910119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (120573119895 + 120573119903)

119909

(13)

Thus we have

119909 + 119903

119910 + 119903

119903

=

1

119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (119895 + 119903)

119909

(14)

Letrsquos state these relations formally in the following theo-rem

Theorem 4 The 119903-Stirling numbers of the second kind withcomplex arguments satisfy the following relations

119909 + 119903

119910 + 119903

119903

=

119909 + 119903 minus 1

119910 + 119903 minus 1

119903

+ (119910 + 119903)

119909 + 119903 minus 1

119910 + 119903

119903

(15)

119909 + 119903

119910 + 119903

119903

=

1

119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (119895 + 119903)

119909

(16)

Remark 5 We can give an alternative proof of this theoremThat is by applying integration by parts on (5) and using thefact that 119890120573119911(119890119911 minus 1)119910minus1 = (119890119911 minus 1)119910 + (119890119911 minus 1)119910minus1 we obtain

119909 + 119903

119910 + 119903

119903

=

(119909 minus 1)

(119910 minus 1)2120587119894

int

H

119890119903119911

(119890119911

minus 1)119910minus1 119889119911

119911119909

+

(119910 + 119903) (119909 minus 1)

1199102120587119894

int

H

119890119903119911

(119890119911

minus 1)119910 119889119911

119911119909

(17)

ISRN Discrete Mathematics 3

This implies (15) immediately On the other hand usingNewtonrsquos Binomial Theorem (5) can be expressed as

119909 + 119903

119910 + 119903

119903

=

119909

119910

1

2120587119894

int

H

119890119903119911

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) 119890120573119911119895119889119911

119911119909+1

=

1

119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895)

119909

2120587119894

int

H

119890(119895+119903)119911

119889119911

119911119909+1

(18)

We know that Hankelrsquos contour integral is a unit for gammafunction over the set of complex numbers that is

(

119894

2120587

int

H

(minus119911)minus119909

119890minus119911

119889119911) Γ (119909) = 1 |119911| lt infin (19)

This implies that

119909

2120587119894

int

H

119890(119895+119903)119911

119889119911

119911119909+1= (119895 + 119903)

119909

(20)

This completes the proof of (16)

The next theorem contains a property for 119909+119903119910+119903 119903which

is analogous to the identity that usually defines Stirling-typenumbers

Theorem 6 The 119903-Stirling numbers of the second kind withcomplex arguments satisfy the following relation

(119905 + 119903)119909

=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(119905)119896 (21)

where (119905)119896= 119905(119905 minus 1) sdot sdot sdot (119905 minus 119896 + 1)

Proof Using (5) with 119910 = 119896 a nonnegative integer we obtain

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(119905)119896=

infin

sum

119896=0

119909

1198962120587119894

int

H

119890119903119911

(119890119911

minus 1)119896 119889119911

119911119909+1 (119905)119896

=

119909

2120587119894

int

H

119890119903119911

infin

sum

119896=0

(

119905

119896) (119890119911

minus 1)119896

119889119911

119911119909+1

=

119909

2120587119894

int

H

119890119903119911

(119890119911

minus 1 + 1)119905 119889119911

119911119909+1

=

119909

2120587119894

int

H

119890119903119911

119890119905119911119889119911

119911119909+1

(22)

Thus by (20) we prove the theorem

Remark 7 It is worth mentioning that this type of propertywas not established for (119903 120573)-Stirling numbers with complex

arguments However replacing 119905 with 120573minus1119905 and 119903 with 119903120573 in(21) we obtain

(120573minus1

119905 +

119903

120573

)

119909

=

infin

sum

119896=0

119909 +

119903

120573

119896 +

119903

120573

119903120573

(120573minus1

119905)119896

(119905 + 119903)119909

=

infin

sum

119896=0

120573119909minus119896

119909 +

119903

120573

119896 +

119903

120573

119903120573

(119905 | 120573)119896

(23)

Using relation (6) we get

(119905 + 119903)119909

=

infin

sum

119896=0

119909

119896

120573119903

(119905 | 120573)119896 (24)

This is the corresponding property for (119903 120573)-Stirling numberswith complex arguments

Remark 8 In a separate paper of the present authors anasymptotic formula for 119903-Stirling numbers of the second kindwith real arguments was established using the method ofChelluri et al [13]

3 119903-Bell Numbers

Using Cauchyrsquos integral formula (2) can be transformed intothe following integral representation of 119861

119899119903

119861119899119903=

119899

2120587119894

int

120574

119890119903119911+(119890

119911minus1)119889119911

119911119899+1 (25)

where the integral contour 120574 is a small contour encircling theorigin Since 119899 is nonnegative in (25) the contour 120574 can bedeformed into a Hankel contour H that starts from minusinfinbelow the negative axis surrounds the origin counterclock-wise and returns to minusinfin in the half plane image119911 gt 0 Weassume that it is at distance le1 from the real axis Now letus consider the following definition for the generalization of119903-Bell numbers 119861

119899119903where 119899 is a complex number

Definition 9 The 119903-Bell numbers 119861119909119903

of complex argument 119909are defined by

119861119909119903=

119909

2120587119894

int

H

119890119903119911+(119890

119911minus1)119889119911

119911119909+1 (26)

where 119903 is complex numbers with real119903 gt 0 and 119909 = Γ(119909 + 1)

Sincereal119903 gt 0 andreal119911 rarr minusinfin clearly the integral in (26)converges for all values of 119909 Moreover 119861

119909119903is a meromorphic

function of 119909 with poles at the nonpositive integersIt can easily be shown that

119861119909119903=

infin

sum

119896=0

1

119896

119896

sum

119895=0

(minus1)119896minus119895

(

119896

119895)

119909

2120587119894

int

H

119890(119895+119903)119911

119911minus(119909+1)

119889119911

(27)

4 ISRN Discrete Mathematics

By (16) and (20) we obtain

119861119909119903=

infin

sum

119896=0

1

119896

119896

sum

119895=0

(minus1)119896minus119895

(

119896

119895) (119895 + 119903)

119909

=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(28)

Hence we have the following theorem

Theorem 10 The 119903-Bell numbers are equal to

119861119909 119903=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(29)

where 119909 and 119903 are complex numbers

By Theorem 10 we have verified that identity in (1) alsoholds for 119861

119909119903with complex argument 119909 This means that

the numbers 119861119909119903

in Definition 9 belong to the family of Bellnumbers

The next theorem asserts that 119861119909119903

also possess a kindof Dobinskirsquos formula which can easily be shown usingDefiniton 9 and the expansion of 119890(119903119911+119890

119911minus1)

Theorem 11 The 119903-Bell numbers are equal to

119861119909 119903=

1

119890

infin

sum

119896=0

(119896 + 119903)119909

119896

(30)

where 119909 and 119903 are complex numbers

Using (26) and (30) we obtain the following corollarywhich is a kind of extension of the integral formula obtainedby Mezo in [8]

Corollary 12 The following integral identity holds

infin

sum

119896=0

(119896 + 119903)119909

119896

=

119909119890

2120587119894

int

H

119890119903119911+(119890

119911minus1)119889119911

119911119909+1 (31)

The 119903-Bell polynomials of Mezo [8] satisfy the recurrencerelation

119861119909 119903(119908) = 119903119861

119909minus1 119903(119908) + 119908119861

119909minus1 119903+1(119908) (32)

where 119909 and 119903 are nonnegative integers This will reduce to

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

(33)

when119908 = 1 Analogous to this relation we have the followingrelation which can easily be shown using (28) (15) and (16)

Theorem 13 The 119903-Bell numbers satisfy the following relation

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(34)

where 119909 and 119903 are complex numbers

Proof Summing up both sides of (15) gives

119861119909 119903= (119903 + 1) 119861

119909minus1 119903+

119909 + 119903 minus 1

119903 minus 1

119903

+

infin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

(35)

By applying (16) we getinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

infin

sum

119896=0

[

[

1

(119896 minus 1)

119896

sum

119895=0

(minus1)119896minus1minus119895

(

119896 minus 1

119895) (120573119895 + 119903)

119909minus1]

]

minus

infin

sum

119896=0

[

[

minus1

(119896minus1)

119896

sum

119895=0

(minus1)119896minus1minus(119895minus1)

(

119896minus1

119895minus1) ((119895minus1)+1+119903)

119909minus1]

]

(36)

Hence we haveinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

119909 + 119903 minus 1

119903 minus 1

119903

minus 119861119909minus1 119903

+ 119861119909minus1 1+119903

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(37)

Substituting this to (35) completes the proof of the theorem

Note that when the parameters 119909 119910 and 119903 are nonneg-ative integers Definition 1 is just equivalent to the integralrepresentation of the 119903-Stirling numbers of the second kindin [1] Hence the value of 119909+119903minus1

119903minus1119903+1

is equal to 0

4 Asymptotic Formula

An asymptotic formula for 119903-Bell numbers 119861119909119903

was firstestablished by Corcino in [7] But the formula holds onlywhen 119909 is a nonnegative integer Here we aim to obtain anasymptotic formula for 119861

119909119903when 119909 is a real number

Using Definition 9

119861119909119903=

119909

2120587119894

int

H

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (38)

To obtain an asymptotic formula we deform the pathH intothe following contour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 where

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number(ii) 1198622is the line segmentreal119911 = 120598 going from 120582+119894(120575minus2120587)

to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

ISRN Discrete Mathematics 5

The new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the counter-

clockwise senseThis idea of deforming the contourH is alsodone in [13] The integrals along 119862

1 1198622 1198624 and 119862

5are seen

to be

119874(

(2 + 120582)119910

(2120587 minus 120575)119909) (39)

It will also be shown that these integrals go to 0 as 119909 rarr infin

provided that 119909minus119910 ge 11990913 To see this we consider1198621 For the

other contours the estimate can be seen similarly We look at1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

le int

1198621

1003816100381610038161003816100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)

119911119909+1

1003816100381610038161003816100381610038161003816100381610038161003816

|119889119911| (40)

Note that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

=10038161003816100381610038161198901199031199111003816100381610038161003816

100381610038161003816100381610038161003816

119890(119890119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890real(119890119911minus1)

(41)

Turning to 119890119911 minus 1

119890119911

minus 1 = 119890real119911

119890119894image119911

minus 1

= [119890real119911 cosimage119911 minus 1] + 119894119890real119911 sinimage119911

(42)

Thus100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890119890real119911 cosimage119911

119890minus1

lt 119890119903real119911

119890119890real119911

119890minus1

(43)

Consider

119903real119911 + 119890real119911

minus 1 lt 119903120582 + 119890120582

minus 1

= 119903120582 + 120582(1 +

120582

2

+

1205822

3

+

1205823

4

+ sdot sdot sdot)

lt 119903120582 + 120582(1 +

120582

2

+

1205822

22+

1205823

23+ sdot sdot sdot)

= 119903120582 + 120582 (

2

2 minus 120582

)

(44)

Choose 120582 lt 1 so that 2 minus 120582 gt 1 Then

119903real119911 + 119890real119911

minus 1 lt 120582 (2 + 119903) (45)

This implies that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

lt 119890120582(2+119903)

(46)

Consequently we obtain1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

lt 119890120582(2+119903)

1

(2120587 minus 120575)119909+1119897 (1198621) (47)

where

119897 (1198621) = lim119905rarrminusinfin

120582 minus 119905 (48)

Since 119897(1198621)(2120587minus120575)

119909+1

rarr 0 as 119909 rarr infin we have the integralalong119862

1goes to 0 as 119909 rarr infin So what remains is the integral

along 1198623 That is

119861119909119903sim

119909

2120587119894

int

1198623

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (49)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Hence by

Laplace method or following the analysis in [7]

119861119909119903sim

119909

2120587119894119877119899int

120572

minus120572

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

sim

119909

2120587119894119877119899int

120598

minus120598

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

(50)

In [7] the integration is along a circle about zero with radius119877 This number 119877 is shown to be the unique solution to

120583

119909 minus 119903120583

= 119890minus120583 (51)

as a function of 120583 (see Lemma 3 [7]) We see that the asymp-totic formula for the 119903-Bell numbers obtained in [7] holdsfor real argument 119909 Thus we have the following asymptoticformula

Theorem 14 The 119903-Bell numbers 119861119909119903

with real arguments 119909and 119903 have the following asymptotic formula

119861119909119903sim 119862radic120587[1 +

119863 minus (21198774

+ 91198773

+ 161198772

+ 6119877 + 2)

24119890119877119877(119877 + 1 + 119903119890

minus119877)3

]

(52)

where

119862 =

119909 exp (119903119877 + 119890119877 minus 1)

120587119877119909[2 (1199032119890119877+ 119877119890119877+ 119903119877)]

12

119863 = (31198773

+ 81198772

minus 6119877 minus 4 minus 2119903119890minus119877

) 119903119890minus119877

(53)

and 119877 is the unique positive solution to

120583119890120583

+ 119903120583 minus 119909 = 0 (54)

as a function of 120583

An asymptotic formula for (119903 120573)-Bell numbers 119866119909119903120573

hasalready been established in [14] However the formula holdsonly when 119909 is a nonnegative integer Here using the samemethod as employed above we can show that this asymptoticformula will also work for the case in which the parameters119909 119903 and 120573 are real numbers

Now for real parameters 119909 119903 and 120573 we have

119866119909119903120573=

119909

2120587119894

int

H

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1 (55)

where the path H can also be deformed into the followingcontour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 such that

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number

6 ISRN Discrete Mathematics

(ii) 1198622is the line segmentreal119911 = 120582 going from 120582 + 119894(120575 minus

2120587) to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

Also the new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the

counterclockwise sense It can easily be shown that along thepath 119862

1

100381610038161003816100381610038161003816

119890119903119911+(119890

120573119911minus1)120573100381610038161003816100381610038161003816

lt 119890120582((2120573)+119903)

(56)

Hence as 119909 goes toinfin100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1

100381610038161003816100381610038161003816100381610038161003816

997888rarr 0 (57)

This can be done similarly along1198622 1198624 and119862

5Thus we have

119866119909119903120573sim

119909

2120587119894

int

1198623

119890119903119911+(119890

120573119911minus1)120573

119911119909+1

119889119911 (58)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Then

119866119909119903120573sim

119909

2120587119894119877

119899int

120572

minus120572

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

sim

119909

2120587119894119877

119899int

120598

minus120598

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

(59)

This implies that the asymptotic formula for the (119903 120573)-Bellnumbers obtained in [14] holds for real argumentThus from[14] we have the following asymptotic formula for (119903 120573)-Bellnumbers 119866

119909119903120573with real arguments 119909 119903 and 120573

119866119909119903120573

sim

11990912

(1 + 112119909) exp (119903119877 + 120573minus1119890120573119877 minus 120573 minus 119909) (120573120573119877 + 119903)119909

[(119909 minus 119903119877) 120573minus1]

12

(120573119877 + 1 + 119903119890minus120573119877)

12

times (1 +

119863 + 119864

119865

)

(60)

where

119863 = (31205732

119877

3

+ 8120573119877

3

+ 3120573119877 + 3 minus 10120573minus1

minus 2119903119890minus120573119877

) 119903119890minus120573119877

119864 = (31205733

minus 51205732

) 119877

4

+ (211205732

minus 30120573)119877

3

+ (39120573 minus 55) 119877

2

+ (24 minus 30120573minus1

) 119877 + (3120573minus1

minus 5120573minus2

)

119865 = 24119877119890120573119877

(120573119877 + 1 + 119903119890minus120573119877

)

3

(61)

and 119877 is the unique positive solution to

120583119890120573120583

+ 119903120583 minus 119909 = 0 (62)as a function of 120583

5 Summary and Recommendation

In this paper we have defined 119903-Stirling numbers of thesecond kind and 119903-Bell numbers with complex argumentsusing the concept of Hankel contour and established someproperties parallel to those of the classical Stirling and Bellnumbers Moreover we have derived an asymptotic formulafor 119903-Bell numbers as well as for (119903 120573)-Bell numbers for realarguments using the method of Chelluri and that of Moserand Wyman

We observe that by employing those methods one canpossibly establish an asymptotic formula for 119903-Stirling num-bers of the second kind with real arguments and conse-quently using relation (6) an asymptotic formula for (119903 120573)-Stirling numbers with real arguments

Acknowledgments

The authors would like to acknowledge the support from theOffice of the Vice Chancellor for Research and Developmentof the University of the Philippines Diliman for this researchproject The authors also wish to thank the referees forreading and evaluating the paper thoroughly

References

[1] A Z Broder ldquoThe r-Stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[2] I Mezo ldquoOn the maximum of r-Stirling numbersrdquo Advances inApplied Mathematics vol 41 no 3 pp 293ndash306 2008

[3] I Mezo ldquoNew properties of r-Stirling seriesrdquoActaMathematicaHungarica vol 119 no 4 pp 341ndash358 2008

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the 119903 120573-Stirling numbersrdquo Ars Combinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] C B Corcino ldquoAn asymptotic formula for the r-Bell numbersrdquoMatimyas Matematika vol 24 no 1 pp 9ndash18 2001

[8] I Mezo ldquoThe r-Bell numbersrdquo Journal of Integer Sequences vol14 no 1 article 1111 2011

[9] R B Corcino M B Montero and C B Corcino ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[10] L Moser and M Wyman ldquoAn asymptotic formula for the Bellnumbersrdquo vol 49 pp 49ndash54 1955

[11] R L Graham D E Knuth and O Patashnik Concrete Math-ematics Addison-Wesley Reading Mass USA 2nd edition1994

[12] P Flajolet and H Prodinger ldquoOn Stirling numbers for complexarguments and Hankel contoursrdquo SIAM Journal on DiscreteMathematics vol 12 no 2 pp 155ndash159 1999

[13] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

ISRN Discrete Mathematics 7

[14] R B Corcino and C B Corcino ldquoOn generalized Bell poly-nomialsrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 623456 21 pages 2011

Submit your manuscripts athttpwwwhindawicom

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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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

2 ISRN Discrete Mathematics

numbers of the second kind may be obtained by taking 120573 =1 Hence using (4) we can define the second-kind 119903-Stirlingnumbers with complex arguments as follows

Definition 1 The 119903-Stirling numbers of the second kind119909+119903

119910+119903 119903of complex arguments 119909 and 119910 are defined by

119909 + 119903

119910 + 119903

119903

=

119909

1199102120587119894

int

H

119890119903119911

(119890119911

minus 1)119910 119889119911

119911119909+1 (5)

where 119903 is complex number with real119903 gt 0 and 119909 = Γ(119909 + 1)and the logarithm involved in the functions (119890119911 minus 1)119910 and 119911119909+1is taken to be the principal branch The Hankel contour Hstarts from minusinfin below the negative axis surrounds the origincounterclockwise and returns tominusinfin in the half planeimage119911 gt 0such that it has a distance 120598 le 1 from the nonpositive real axis

Remark 2 Since real119903 gt 0 and real119911 rarr minusinfin the integral in (5)converges for all values of 119909 and 119910 Also 119909+119903119910+119903

119903is a mero-

morphic function of 119909 (for any fixed 119910) with poles at thenonpositive integers and it is entire as a function of 119910 (forany fixed 119909 not a negative integer)

Remark 3 By the change of variable on the integral in (4) say119908 = 120573119911 we can express 119909119910

120573119903in terms of 119909+119903119910+119903

119903as follows

119909

119910

120573119903

= 120573119909minus119910

119909 +

119903

120573

119910 +

119903

120573

119903120573

(6)

Because of relation (6) every property of (119903 120573)-Stirlingnumbers with complex arguments will have a correspondingproperty for 119903-Stirling numbers with complex arguments andvice versa For instance the (119903 120573)-Stirling numbers in [9]satisfy the following relation

119909

119910

120573 119903

=

119909 minus 1

119910 minus 1

120573119903

+ (120573119910 + 119903)

119909 minus 1

119910

120573119903

(7)

Replacing 119903 with 120573119903 we get

119909

119910

120573 120573119903

=

119909 minus 1

119910 minus 1

120573120573119903

+ (120573119910 + 120573119903)

119909 minus 1

119910

120573120573119903

(8)

Using (6) we obtain

120573119909minus119910

119909 +

120573119903

120573

119910 +

120573119903

120573

120573119903120573

= 120573119909minus119910

119909 + (

120573119903

120573

) minus 1

119910 + (

120573119903

120573

) minus 1

120573119903120573

+ 120573119909minus1minus119910

(120573119910 + 120573119903)

119909 + (

120573119903

120573

) minus 1

119910 + 120573119903

120573119903120573

(9)

Thus we have

119909 + 119903

119910 + 119903

119903

=

119909 + 119903 minus 1

119910 + 119903 minus 1

119903

+ (119910 + 119903)

119909 + 119903 minus 1

119910 + 119903

119903

(10)

On the other hand the (119903 120573)-Stirling numbers in [9]satisfy the relation

119909

119910

120573 119903

=

1

120573119910119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (120573119895 + 119903)

119909

(11)

Again replacing 119903 with 120573119903 we obtain

119909

119910

120573120573119903

=

1

120573119910119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (120573119895 + 120573119903)

119909

(12)

By (6) we get

120573119909minus119910

119909 +

120573119903

120573

119910 +

120573119903

120573

120573119903120573

=

1

120573119910119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (120573119895 + 120573119903)

119909

(13)

Thus we have

119909 + 119903

119910 + 119903

119903

=

1

119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (119895 + 119903)

119909

(14)

Letrsquos state these relations formally in the following theo-rem

Theorem 4 The 119903-Stirling numbers of the second kind withcomplex arguments satisfy the following relations

119909 + 119903

119910 + 119903

119903

=

119909 + 119903 minus 1

119910 + 119903 minus 1

119903

+ (119910 + 119903)

119909 + 119903 minus 1

119910 + 119903

119903

(15)

119909 + 119903

119910 + 119903

119903

=

1

119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) (119895 + 119903)

119909

(16)

Remark 5 We can give an alternative proof of this theoremThat is by applying integration by parts on (5) and using thefact that 119890120573119911(119890119911 minus 1)119910minus1 = (119890119911 minus 1)119910 + (119890119911 minus 1)119910minus1 we obtain

119909 + 119903

119910 + 119903

119903

=

(119909 minus 1)

(119910 minus 1)2120587119894

int

H

119890119903119911

(119890119911

minus 1)119910minus1 119889119911

119911119909

+

(119910 + 119903) (119909 minus 1)

1199102120587119894

int

H

119890119903119911

(119890119911

minus 1)119910 119889119911

119911119909

(17)

ISRN Discrete Mathematics 3

This implies (15) immediately On the other hand usingNewtonrsquos Binomial Theorem (5) can be expressed as

119909 + 119903

119910 + 119903

119903

=

119909

119910

1

2120587119894

int

H

119890119903119911

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) 119890120573119911119895119889119911

119911119909+1

=

1

119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895)

119909

2120587119894

int

H

119890(119895+119903)119911

119889119911

119911119909+1

(18)

We know that Hankelrsquos contour integral is a unit for gammafunction over the set of complex numbers that is

(

119894

2120587

int

H

(minus119911)minus119909

119890minus119911

119889119911) Γ (119909) = 1 |119911| lt infin (19)

This implies that

119909

2120587119894

int

H

119890(119895+119903)119911

119889119911

119911119909+1= (119895 + 119903)

119909

(20)

This completes the proof of (16)

The next theorem contains a property for 119909+119903119910+119903 119903which

is analogous to the identity that usually defines Stirling-typenumbers

Theorem 6 The 119903-Stirling numbers of the second kind withcomplex arguments satisfy the following relation

(119905 + 119903)119909

=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(119905)119896 (21)

where (119905)119896= 119905(119905 minus 1) sdot sdot sdot (119905 minus 119896 + 1)

Proof Using (5) with 119910 = 119896 a nonnegative integer we obtain

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(119905)119896=

infin

sum

119896=0

119909

1198962120587119894

int

H

119890119903119911

(119890119911

minus 1)119896 119889119911

119911119909+1 (119905)119896

=

119909

2120587119894

int

H

119890119903119911

infin

sum

119896=0

(

119905

119896) (119890119911

minus 1)119896

119889119911

119911119909+1

=

119909

2120587119894

int

H

119890119903119911

(119890119911

minus 1 + 1)119905 119889119911

119911119909+1

=

119909

2120587119894

int

H

119890119903119911

119890119905119911119889119911

119911119909+1

(22)

Thus by (20) we prove the theorem

Remark 7 It is worth mentioning that this type of propertywas not established for (119903 120573)-Stirling numbers with complex

arguments However replacing 119905 with 120573minus1119905 and 119903 with 119903120573 in(21) we obtain

(120573minus1

119905 +

119903

120573

)

119909

=

infin

sum

119896=0

119909 +

119903

120573

119896 +

119903

120573

119903120573

(120573minus1

119905)119896

(119905 + 119903)119909

=

infin

sum

119896=0

120573119909minus119896

119909 +

119903

120573

119896 +

119903

120573

119903120573

(119905 | 120573)119896

(23)

Using relation (6) we get

(119905 + 119903)119909

=

infin

sum

119896=0

119909

119896

120573119903

(119905 | 120573)119896 (24)

This is the corresponding property for (119903 120573)-Stirling numberswith complex arguments

Remark 8 In a separate paper of the present authors anasymptotic formula for 119903-Stirling numbers of the second kindwith real arguments was established using the method ofChelluri et al [13]

3 119903-Bell Numbers

Using Cauchyrsquos integral formula (2) can be transformed intothe following integral representation of 119861

119899119903

119861119899119903=

119899

2120587119894

int

120574

119890119903119911+(119890

119911minus1)119889119911

119911119899+1 (25)

where the integral contour 120574 is a small contour encircling theorigin Since 119899 is nonnegative in (25) the contour 120574 can bedeformed into a Hankel contour H that starts from minusinfinbelow the negative axis surrounds the origin counterclock-wise and returns to minusinfin in the half plane image119911 gt 0 Weassume that it is at distance le1 from the real axis Now letus consider the following definition for the generalization of119903-Bell numbers 119861

119899119903where 119899 is a complex number

Definition 9 The 119903-Bell numbers 119861119909119903

of complex argument 119909are defined by

119861119909119903=

119909

2120587119894

int

H

119890119903119911+(119890

119911minus1)119889119911

119911119909+1 (26)

where 119903 is complex numbers with real119903 gt 0 and 119909 = Γ(119909 + 1)

Sincereal119903 gt 0 andreal119911 rarr minusinfin clearly the integral in (26)converges for all values of 119909 Moreover 119861

119909119903is a meromorphic

function of 119909 with poles at the nonpositive integersIt can easily be shown that

119861119909119903=

infin

sum

119896=0

1

119896

119896

sum

119895=0

(minus1)119896minus119895

(

119896

119895)

119909

2120587119894

int

H

119890(119895+119903)119911

119911minus(119909+1)

119889119911

(27)

4 ISRN Discrete Mathematics

By (16) and (20) we obtain

119861119909119903=

infin

sum

119896=0

1

119896

119896

sum

119895=0

(minus1)119896minus119895

(

119896

119895) (119895 + 119903)

119909

=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(28)

Hence we have the following theorem

Theorem 10 The 119903-Bell numbers are equal to

119861119909 119903=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(29)

where 119909 and 119903 are complex numbers

By Theorem 10 we have verified that identity in (1) alsoholds for 119861

119909119903with complex argument 119909 This means that

the numbers 119861119909119903

in Definition 9 belong to the family of Bellnumbers

The next theorem asserts that 119861119909119903

also possess a kindof Dobinskirsquos formula which can easily be shown usingDefiniton 9 and the expansion of 119890(119903119911+119890

119911minus1)

Theorem 11 The 119903-Bell numbers are equal to

119861119909 119903=

1

119890

infin

sum

119896=0

(119896 + 119903)119909

119896

(30)

where 119909 and 119903 are complex numbers

Using (26) and (30) we obtain the following corollarywhich is a kind of extension of the integral formula obtainedby Mezo in [8]

Corollary 12 The following integral identity holds

infin

sum

119896=0

(119896 + 119903)119909

119896

=

119909119890

2120587119894

int

H

119890119903119911+(119890

119911minus1)119889119911

119911119909+1 (31)

The 119903-Bell polynomials of Mezo [8] satisfy the recurrencerelation

119861119909 119903(119908) = 119903119861

119909minus1 119903(119908) + 119908119861

119909minus1 119903+1(119908) (32)

where 119909 and 119903 are nonnegative integers This will reduce to

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

(33)

when119908 = 1 Analogous to this relation we have the followingrelation which can easily be shown using (28) (15) and (16)

Theorem 13 The 119903-Bell numbers satisfy the following relation

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(34)

where 119909 and 119903 are complex numbers

Proof Summing up both sides of (15) gives

119861119909 119903= (119903 + 1) 119861

119909minus1 119903+

119909 + 119903 minus 1

119903 minus 1

119903

+

infin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

(35)

By applying (16) we getinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

infin

sum

119896=0

[

[

1

(119896 minus 1)

119896

sum

119895=0

(minus1)119896minus1minus119895

(

119896 minus 1

119895) (120573119895 + 119903)

119909minus1]

]

minus

infin

sum

119896=0

[

[

minus1

(119896minus1)

119896

sum

119895=0

(minus1)119896minus1minus(119895minus1)

(

119896minus1

119895minus1) ((119895minus1)+1+119903)

119909minus1]

]

(36)

Hence we haveinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

119909 + 119903 minus 1

119903 minus 1

119903

minus 119861119909minus1 119903

+ 119861119909minus1 1+119903

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(37)

Substituting this to (35) completes the proof of the theorem

Note that when the parameters 119909 119910 and 119903 are nonneg-ative integers Definition 1 is just equivalent to the integralrepresentation of the 119903-Stirling numbers of the second kindin [1] Hence the value of 119909+119903minus1

119903minus1119903+1

is equal to 0

4 Asymptotic Formula

An asymptotic formula for 119903-Bell numbers 119861119909119903

was firstestablished by Corcino in [7] But the formula holds onlywhen 119909 is a nonnegative integer Here we aim to obtain anasymptotic formula for 119861

119909119903when 119909 is a real number

Using Definition 9

119861119909119903=

119909

2120587119894

int

H

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (38)

To obtain an asymptotic formula we deform the pathH intothe following contour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 where

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number(ii) 1198622is the line segmentreal119911 = 120598 going from 120582+119894(120575minus2120587)

to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

ISRN Discrete Mathematics 5

The new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the counter-

clockwise senseThis idea of deforming the contourH is alsodone in [13] The integrals along 119862

1 1198622 1198624 and 119862

5are seen

to be

119874(

(2 + 120582)119910

(2120587 minus 120575)119909) (39)

It will also be shown that these integrals go to 0 as 119909 rarr infin

provided that 119909minus119910 ge 11990913 To see this we consider1198621 For the

other contours the estimate can be seen similarly We look at1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

le int

1198621

1003816100381610038161003816100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)

119911119909+1

1003816100381610038161003816100381610038161003816100381610038161003816

|119889119911| (40)

Note that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

=10038161003816100381610038161198901199031199111003816100381610038161003816

100381610038161003816100381610038161003816

119890(119890119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890real(119890119911minus1)

(41)

Turning to 119890119911 minus 1

119890119911

minus 1 = 119890real119911

119890119894image119911

minus 1

= [119890real119911 cosimage119911 minus 1] + 119894119890real119911 sinimage119911

(42)

Thus100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890119890real119911 cosimage119911

119890minus1

lt 119890119903real119911

119890119890real119911

119890minus1

(43)

Consider

119903real119911 + 119890real119911

minus 1 lt 119903120582 + 119890120582

minus 1

= 119903120582 + 120582(1 +

120582

2

+

1205822

3

+

1205823

4

+ sdot sdot sdot)

lt 119903120582 + 120582(1 +

120582

2

+

1205822

22+

1205823

23+ sdot sdot sdot)

= 119903120582 + 120582 (

2

2 minus 120582

)

(44)

Choose 120582 lt 1 so that 2 minus 120582 gt 1 Then

119903real119911 + 119890real119911

minus 1 lt 120582 (2 + 119903) (45)

This implies that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

lt 119890120582(2+119903)

(46)

Consequently we obtain1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

lt 119890120582(2+119903)

1

(2120587 minus 120575)119909+1119897 (1198621) (47)

where

119897 (1198621) = lim119905rarrminusinfin

120582 minus 119905 (48)

Since 119897(1198621)(2120587minus120575)

119909+1

rarr 0 as 119909 rarr infin we have the integralalong119862

1goes to 0 as 119909 rarr infin So what remains is the integral

along 1198623 That is

119861119909119903sim

119909

2120587119894

int

1198623

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (49)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Hence by

Laplace method or following the analysis in [7]

119861119909119903sim

119909

2120587119894119877119899int

120572

minus120572

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

sim

119909

2120587119894119877119899int

120598

minus120598

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

(50)

In [7] the integration is along a circle about zero with radius119877 This number 119877 is shown to be the unique solution to

120583

119909 minus 119903120583

= 119890minus120583 (51)

as a function of 120583 (see Lemma 3 [7]) We see that the asymp-totic formula for the 119903-Bell numbers obtained in [7] holdsfor real argument 119909 Thus we have the following asymptoticformula

Theorem 14 The 119903-Bell numbers 119861119909119903

with real arguments 119909and 119903 have the following asymptotic formula

119861119909119903sim 119862radic120587[1 +

119863 minus (21198774

+ 91198773

+ 161198772

+ 6119877 + 2)

24119890119877119877(119877 + 1 + 119903119890

minus119877)3

]

(52)

where

119862 =

119909 exp (119903119877 + 119890119877 minus 1)

120587119877119909[2 (1199032119890119877+ 119877119890119877+ 119903119877)]

12

119863 = (31198773

+ 81198772

minus 6119877 minus 4 minus 2119903119890minus119877

) 119903119890minus119877

(53)

and 119877 is the unique positive solution to

120583119890120583

+ 119903120583 minus 119909 = 0 (54)

as a function of 120583

An asymptotic formula for (119903 120573)-Bell numbers 119866119909119903120573

hasalready been established in [14] However the formula holdsonly when 119909 is a nonnegative integer Here using the samemethod as employed above we can show that this asymptoticformula will also work for the case in which the parameters119909 119903 and 120573 are real numbers

Now for real parameters 119909 119903 and 120573 we have

119866119909119903120573=

119909

2120587119894

int

H

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1 (55)

where the path H can also be deformed into the followingcontour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 such that

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number

6 ISRN Discrete Mathematics

(ii) 1198622is the line segmentreal119911 = 120582 going from 120582 + 119894(120575 minus

2120587) to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

Also the new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the

counterclockwise sense It can easily be shown that along thepath 119862

1

100381610038161003816100381610038161003816

119890119903119911+(119890

120573119911minus1)120573100381610038161003816100381610038161003816

lt 119890120582((2120573)+119903)

(56)

Hence as 119909 goes toinfin100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1

100381610038161003816100381610038161003816100381610038161003816

997888rarr 0 (57)

This can be done similarly along1198622 1198624 and119862

5Thus we have

119866119909119903120573sim

119909

2120587119894

int

1198623

119890119903119911+(119890

120573119911minus1)120573

119911119909+1

119889119911 (58)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Then

119866119909119903120573sim

119909

2120587119894119877

119899int

120572

minus120572

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

sim

119909

2120587119894119877

119899int

120598

minus120598

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

(59)

This implies that the asymptotic formula for the (119903 120573)-Bellnumbers obtained in [14] holds for real argumentThus from[14] we have the following asymptotic formula for (119903 120573)-Bellnumbers 119866

119909119903120573with real arguments 119909 119903 and 120573

119866119909119903120573

sim

11990912

(1 + 112119909) exp (119903119877 + 120573minus1119890120573119877 minus 120573 minus 119909) (120573120573119877 + 119903)119909

[(119909 minus 119903119877) 120573minus1]

12

(120573119877 + 1 + 119903119890minus120573119877)

12

times (1 +

119863 + 119864

119865

)

(60)

where

119863 = (31205732

119877

3

+ 8120573119877

3

+ 3120573119877 + 3 minus 10120573minus1

minus 2119903119890minus120573119877

) 119903119890minus120573119877

119864 = (31205733

minus 51205732

) 119877

4

+ (211205732

minus 30120573)119877

3

+ (39120573 minus 55) 119877

2

+ (24 minus 30120573minus1

) 119877 + (3120573minus1

minus 5120573minus2

)

119865 = 24119877119890120573119877

(120573119877 + 1 + 119903119890minus120573119877

)

3

(61)

and 119877 is the unique positive solution to

120583119890120573120583

+ 119903120583 minus 119909 = 0 (62)as a function of 120583

5 Summary and Recommendation

In this paper we have defined 119903-Stirling numbers of thesecond kind and 119903-Bell numbers with complex argumentsusing the concept of Hankel contour and established someproperties parallel to those of the classical Stirling and Bellnumbers Moreover we have derived an asymptotic formulafor 119903-Bell numbers as well as for (119903 120573)-Bell numbers for realarguments using the method of Chelluri and that of Moserand Wyman

We observe that by employing those methods one canpossibly establish an asymptotic formula for 119903-Stirling num-bers of the second kind with real arguments and conse-quently using relation (6) an asymptotic formula for (119903 120573)-Stirling numbers with real arguments

Acknowledgments

The authors would like to acknowledge the support from theOffice of the Vice Chancellor for Research and Developmentof the University of the Philippines Diliman for this researchproject The authors also wish to thank the referees forreading and evaluating the paper thoroughly

References

[1] A Z Broder ldquoThe r-Stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[2] I Mezo ldquoOn the maximum of r-Stirling numbersrdquo Advances inApplied Mathematics vol 41 no 3 pp 293ndash306 2008

[3] I Mezo ldquoNew properties of r-Stirling seriesrdquoActaMathematicaHungarica vol 119 no 4 pp 341ndash358 2008

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the 119903 120573-Stirling numbersrdquo Ars Combinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] C B Corcino ldquoAn asymptotic formula for the r-Bell numbersrdquoMatimyas Matematika vol 24 no 1 pp 9ndash18 2001

[8] I Mezo ldquoThe r-Bell numbersrdquo Journal of Integer Sequences vol14 no 1 article 1111 2011

[9] R B Corcino M B Montero and C B Corcino ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[10] L Moser and M Wyman ldquoAn asymptotic formula for the Bellnumbersrdquo vol 49 pp 49ndash54 1955

[11] R L Graham D E Knuth and O Patashnik Concrete Math-ematics Addison-Wesley Reading Mass USA 2nd edition1994

[12] P Flajolet and H Prodinger ldquoOn Stirling numbers for complexarguments and Hankel contoursrdquo SIAM Journal on DiscreteMathematics vol 12 no 2 pp 155ndash159 1999

[13] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

ISRN Discrete Mathematics 7

[14] R B Corcino and C B Corcino ldquoOn generalized Bell poly-nomialsrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 623456 21 pages 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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OptimizationJournal of

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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

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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

ISRN Discrete Mathematics 3

This implies (15) immediately On the other hand usingNewtonrsquos Binomial Theorem (5) can be expressed as

119909 + 119903

119910 + 119903

119903

=

119909

119910

1

2120587119894

int

H

119890119903119911

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895) 119890120573119911119895119889119911

119911119909+1

=

1

119910

infin

sum

119895=0

(minus1)119910minus119895

(

119910

119895)

119909

2120587119894

int

H

119890(119895+119903)119911

119889119911

119911119909+1

(18)

We know that Hankelrsquos contour integral is a unit for gammafunction over the set of complex numbers that is

(

119894

2120587

int

H

(minus119911)minus119909

119890minus119911

119889119911) Γ (119909) = 1 |119911| lt infin (19)

This implies that

119909

2120587119894

int

H

119890(119895+119903)119911

119889119911

119911119909+1= (119895 + 119903)

119909

(20)

This completes the proof of (16)

The next theorem contains a property for 119909+119903119910+119903 119903which

is analogous to the identity that usually defines Stirling-typenumbers

Theorem 6 The 119903-Stirling numbers of the second kind withcomplex arguments satisfy the following relation

(119905 + 119903)119909

=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(119905)119896 (21)

where (119905)119896= 119905(119905 minus 1) sdot sdot sdot (119905 minus 119896 + 1)

Proof Using (5) with 119910 = 119896 a nonnegative integer we obtain

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(119905)119896=

infin

sum

119896=0

119909

1198962120587119894

int

H

119890119903119911

(119890119911

minus 1)119896 119889119911

119911119909+1 (119905)119896

=

119909

2120587119894

int

H

119890119903119911

infin

sum

119896=0

(

119905

119896) (119890119911

minus 1)119896

119889119911

119911119909+1

=

119909

2120587119894

int

H

119890119903119911

(119890119911

minus 1 + 1)119905 119889119911

119911119909+1

=

119909

2120587119894

int

H

119890119903119911

119890119905119911119889119911

119911119909+1

(22)

Thus by (20) we prove the theorem

Remark 7 It is worth mentioning that this type of propertywas not established for (119903 120573)-Stirling numbers with complex

arguments However replacing 119905 with 120573minus1119905 and 119903 with 119903120573 in(21) we obtain

(120573minus1

119905 +

119903

120573

)

119909

=

infin

sum

119896=0

119909 +

119903

120573

119896 +

119903

120573

119903120573

(120573minus1

119905)119896

(119905 + 119903)119909

=

infin

sum

119896=0

120573119909minus119896

119909 +

119903

120573

119896 +

119903

120573

119903120573

(119905 | 120573)119896

(23)

Using relation (6) we get

(119905 + 119903)119909

=

infin

sum

119896=0

119909

119896

120573119903

(119905 | 120573)119896 (24)

This is the corresponding property for (119903 120573)-Stirling numberswith complex arguments

Remark 8 In a separate paper of the present authors anasymptotic formula for 119903-Stirling numbers of the second kindwith real arguments was established using the method ofChelluri et al [13]

3 119903-Bell Numbers

Using Cauchyrsquos integral formula (2) can be transformed intothe following integral representation of 119861

119899119903

119861119899119903=

119899

2120587119894

int

120574

119890119903119911+(119890

119911minus1)119889119911

119911119899+1 (25)

where the integral contour 120574 is a small contour encircling theorigin Since 119899 is nonnegative in (25) the contour 120574 can bedeformed into a Hankel contour H that starts from minusinfinbelow the negative axis surrounds the origin counterclock-wise and returns to minusinfin in the half plane image119911 gt 0 Weassume that it is at distance le1 from the real axis Now letus consider the following definition for the generalization of119903-Bell numbers 119861

119899119903where 119899 is a complex number

Definition 9 The 119903-Bell numbers 119861119909119903

of complex argument 119909are defined by

119861119909119903=

119909

2120587119894

int

H

119890119903119911+(119890

119911minus1)119889119911

119911119909+1 (26)

where 119903 is complex numbers with real119903 gt 0 and 119909 = Γ(119909 + 1)

Sincereal119903 gt 0 andreal119911 rarr minusinfin clearly the integral in (26)converges for all values of 119909 Moreover 119861

119909119903is a meromorphic

function of 119909 with poles at the nonpositive integersIt can easily be shown that

119861119909119903=

infin

sum

119896=0

1

119896

119896

sum

119895=0

(minus1)119896minus119895

(

119896

119895)

119909

2120587119894

int

H

119890(119895+119903)119911

119911minus(119909+1)

119889119911

(27)

4 ISRN Discrete Mathematics

By (16) and (20) we obtain

119861119909119903=

infin

sum

119896=0

1

119896

119896

sum

119895=0

(minus1)119896minus119895

(

119896

119895) (119895 + 119903)

119909

=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(28)

Hence we have the following theorem

Theorem 10 The 119903-Bell numbers are equal to

119861119909 119903=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(29)

where 119909 and 119903 are complex numbers

By Theorem 10 we have verified that identity in (1) alsoholds for 119861

119909119903with complex argument 119909 This means that

the numbers 119861119909119903

in Definition 9 belong to the family of Bellnumbers

The next theorem asserts that 119861119909119903

also possess a kindof Dobinskirsquos formula which can easily be shown usingDefiniton 9 and the expansion of 119890(119903119911+119890

119911minus1)

Theorem 11 The 119903-Bell numbers are equal to

119861119909 119903=

1

119890

infin

sum

119896=0

(119896 + 119903)119909

119896

(30)

where 119909 and 119903 are complex numbers

Using (26) and (30) we obtain the following corollarywhich is a kind of extension of the integral formula obtainedby Mezo in [8]

Corollary 12 The following integral identity holds

infin

sum

119896=0

(119896 + 119903)119909

119896

=

119909119890

2120587119894

int

H

119890119903119911+(119890

119911minus1)119889119911

119911119909+1 (31)

The 119903-Bell polynomials of Mezo [8] satisfy the recurrencerelation

119861119909 119903(119908) = 119903119861

119909minus1 119903(119908) + 119908119861

119909minus1 119903+1(119908) (32)

where 119909 and 119903 are nonnegative integers This will reduce to

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

(33)

when119908 = 1 Analogous to this relation we have the followingrelation which can easily be shown using (28) (15) and (16)

Theorem 13 The 119903-Bell numbers satisfy the following relation

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(34)

where 119909 and 119903 are complex numbers

Proof Summing up both sides of (15) gives

119861119909 119903= (119903 + 1) 119861

119909minus1 119903+

119909 + 119903 minus 1

119903 minus 1

119903

+

infin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

(35)

By applying (16) we getinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

infin

sum

119896=0

[

[

1

(119896 minus 1)

119896

sum

119895=0

(minus1)119896minus1minus119895

(

119896 minus 1

119895) (120573119895 + 119903)

119909minus1]

]

minus

infin

sum

119896=0

[

[

minus1

(119896minus1)

119896

sum

119895=0

(minus1)119896minus1minus(119895minus1)

(

119896minus1

119895minus1) ((119895minus1)+1+119903)

119909minus1]

]

(36)

Hence we haveinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

119909 + 119903 minus 1

119903 minus 1

119903

minus 119861119909minus1 119903

+ 119861119909minus1 1+119903

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(37)

Substituting this to (35) completes the proof of the theorem

Note that when the parameters 119909 119910 and 119903 are nonneg-ative integers Definition 1 is just equivalent to the integralrepresentation of the 119903-Stirling numbers of the second kindin [1] Hence the value of 119909+119903minus1

119903minus1119903+1

is equal to 0

4 Asymptotic Formula

An asymptotic formula for 119903-Bell numbers 119861119909119903

was firstestablished by Corcino in [7] But the formula holds onlywhen 119909 is a nonnegative integer Here we aim to obtain anasymptotic formula for 119861

119909119903when 119909 is a real number

Using Definition 9

119861119909119903=

119909

2120587119894

int

H

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (38)

To obtain an asymptotic formula we deform the pathH intothe following contour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 where

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number(ii) 1198622is the line segmentreal119911 = 120598 going from 120582+119894(120575minus2120587)

to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

ISRN Discrete Mathematics 5

The new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the counter-

clockwise senseThis idea of deforming the contourH is alsodone in [13] The integrals along 119862

1 1198622 1198624 and 119862

5are seen

to be

119874(

(2 + 120582)119910

(2120587 minus 120575)119909) (39)

It will also be shown that these integrals go to 0 as 119909 rarr infin

provided that 119909minus119910 ge 11990913 To see this we consider1198621 For the

other contours the estimate can be seen similarly We look at1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

le int

1198621

1003816100381610038161003816100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)

119911119909+1

1003816100381610038161003816100381610038161003816100381610038161003816

|119889119911| (40)

Note that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

=10038161003816100381610038161198901199031199111003816100381610038161003816

100381610038161003816100381610038161003816

119890(119890119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890real(119890119911minus1)

(41)

Turning to 119890119911 minus 1

119890119911

minus 1 = 119890real119911

119890119894image119911

minus 1

= [119890real119911 cosimage119911 minus 1] + 119894119890real119911 sinimage119911

(42)

Thus100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890119890real119911 cosimage119911

119890minus1

lt 119890119903real119911

119890119890real119911

119890minus1

(43)

Consider

119903real119911 + 119890real119911

minus 1 lt 119903120582 + 119890120582

minus 1

= 119903120582 + 120582(1 +

120582

2

+

1205822

3

+

1205823

4

+ sdot sdot sdot)

lt 119903120582 + 120582(1 +

120582

2

+

1205822

22+

1205823

23+ sdot sdot sdot)

= 119903120582 + 120582 (

2

2 minus 120582

)

(44)

Choose 120582 lt 1 so that 2 minus 120582 gt 1 Then

119903real119911 + 119890real119911

minus 1 lt 120582 (2 + 119903) (45)

This implies that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

lt 119890120582(2+119903)

(46)

Consequently we obtain1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

lt 119890120582(2+119903)

1

(2120587 minus 120575)119909+1119897 (1198621) (47)

where

119897 (1198621) = lim119905rarrminusinfin

120582 minus 119905 (48)

Since 119897(1198621)(2120587minus120575)

119909+1

rarr 0 as 119909 rarr infin we have the integralalong119862

1goes to 0 as 119909 rarr infin So what remains is the integral

along 1198623 That is

119861119909119903sim

119909

2120587119894

int

1198623

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (49)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Hence by

Laplace method or following the analysis in [7]

119861119909119903sim

119909

2120587119894119877119899int

120572

minus120572

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

sim

119909

2120587119894119877119899int

120598

minus120598

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

(50)

In [7] the integration is along a circle about zero with radius119877 This number 119877 is shown to be the unique solution to

120583

119909 minus 119903120583

= 119890minus120583 (51)

as a function of 120583 (see Lemma 3 [7]) We see that the asymp-totic formula for the 119903-Bell numbers obtained in [7] holdsfor real argument 119909 Thus we have the following asymptoticformula

Theorem 14 The 119903-Bell numbers 119861119909119903

with real arguments 119909and 119903 have the following asymptotic formula

119861119909119903sim 119862radic120587[1 +

119863 minus (21198774

+ 91198773

+ 161198772

+ 6119877 + 2)

24119890119877119877(119877 + 1 + 119903119890

minus119877)3

]

(52)

where

119862 =

119909 exp (119903119877 + 119890119877 minus 1)

120587119877119909[2 (1199032119890119877+ 119877119890119877+ 119903119877)]

12

119863 = (31198773

+ 81198772

minus 6119877 minus 4 minus 2119903119890minus119877

) 119903119890minus119877

(53)

and 119877 is the unique positive solution to

120583119890120583

+ 119903120583 minus 119909 = 0 (54)

as a function of 120583

An asymptotic formula for (119903 120573)-Bell numbers 119866119909119903120573

hasalready been established in [14] However the formula holdsonly when 119909 is a nonnegative integer Here using the samemethod as employed above we can show that this asymptoticformula will also work for the case in which the parameters119909 119903 and 120573 are real numbers

Now for real parameters 119909 119903 and 120573 we have

119866119909119903120573=

119909

2120587119894

int

H

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1 (55)

where the path H can also be deformed into the followingcontour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 such that

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number

6 ISRN Discrete Mathematics

(ii) 1198622is the line segmentreal119911 = 120582 going from 120582 + 119894(120575 minus

2120587) to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

Also the new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the

counterclockwise sense It can easily be shown that along thepath 119862

1

100381610038161003816100381610038161003816

119890119903119911+(119890

120573119911minus1)120573100381610038161003816100381610038161003816

lt 119890120582((2120573)+119903)

(56)

Hence as 119909 goes toinfin100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1

100381610038161003816100381610038161003816100381610038161003816

997888rarr 0 (57)

This can be done similarly along1198622 1198624 and119862

5Thus we have

119866119909119903120573sim

119909

2120587119894

int

1198623

119890119903119911+(119890

120573119911minus1)120573

119911119909+1

119889119911 (58)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Then

119866119909119903120573sim

119909

2120587119894119877

119899int

120572

minus120572

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

sim

119909

2120587119894119877

119899int

120598

minus120598

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

(59)

This implies that the asymptotic formula for the (119903 120573)-Bellnumbers obtained in [14] holds for real argumentThus from[14] we have the following asymptotic formula for (119903 120573)-Bellnumbers 119866

119909119903120573with real arguments 119909 119903 and 120573

119866119909119903120573

sim

11990912

(1 + 112119909) exp (119903119877 + 120573minus1119890120573119877 minus 120573 minus 119909) (120573120573119877 + 119903)119909

[(119909 minus 119903119877) 120573minus1]

12

(120573119877 + 1 + 119903119890minus120573119877)

12

times (1 +

119863 + 119864

119865

)

(60)

where

119863 = (31205732

119877

3

+ 8120573119877

3

+ 3120573119877 + 3 minus 10120573minus1

minus 2119903119890minus120573119877

) 119903119890minus120573119877

119864 = (31205733

minus 51205732

) 119877

4

+ (211205732

minus 30120573)119877

3

+ (39120573 minus 55) 119877

2

+ (24 minus 30120573minus1

) 119877 + (3120573minus1

minus 5120573minus2

)

119865 = 24119877119890120573119877

(120573119877 + 1 + 119903119890minus120573119877

)

3

(61)

and 119877 is the unique positive solution to

120583119890120573120583

+ 119903120583 minus 119909 = 0 (62)as a function of 120583

5 Summary and Recommendation

In this paper we have defined 119903-Stirling numbers of thesecond kind and 119903-Bell numbers with complex argumentsusing the concept of Hankel contour and established someproperties parallel to those of the classical Stirling and Bellnumbers Moreover we have derived an asymptotic formulafor 119903-Bell numbers as well as for (119903 120573)-Bell numbers for realarguments using the method of Chelluri and that of Moserand Wyman

We observe that by employing those methods one canpossibly establish an asymptotic formula for 119903-Stirling num-bers of the second kind with real arguments and conse-quently using relation (6) an asymptotic formula for (119903 120573)-Stirling numbers with real arguments

Acknowledgments

The authors would like to acknowledge the support from theOffice of the Vice Chancellor for Research and Developmentof the University of the Philippines Diliman for this researchproject The authors also wish to thank the referees forreading and evaluating the paper thoroughly

References

[1] A Z Broder ldquoThe r-Stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[2] I Mezo ldquoOn the maximum of r-Stirling numbersrdquo Advances inApplied Mathematics vol 41 no 3 pp 293ndash306 2008

[3] I Mezo ldquoNew properties of r-Stirling seriesrdquoActaMathematicaHungarica vol 119 no 4 pp 341ndash358 2008

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the 119903 120573-Stirling numbersrdquo Ars Combinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] C B Corcino ldquoAn asymptotic formula for the r-Bell numbersrdquoMatimyas Matematika vol 24 no 1 pp 9ndash18 2001

[8] I Mezo ldquoThe r-Bell numbersrdquo Journal of Integer Sequences vol14 no 1 article 1111 2011

[9] R B Corcino M B Montero and C B Corcino ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[10] L Moser and M Wyman ldquoAn asymptotic formula for the Bellnumbersrdquo vol 49 pp 49ndash54 1955

[11] R L Graham D E Knuth and O Patashnik Concrete Math-ematics Addison-Wesley Reading Mass USA 2nd edition1994

[12] P Flajolet and H Prodinger ldquoOn Stirling numbers for complexarguments and Hankel contoursrdquo SIAM Journal on DiscreteMathematics vol 12 no 2 pp 155ndash159 1999

[13] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

ISRN Discrete Mathematics 7

[14] R B Corcino and C B Corcino ldquoOn generalized Bell poly-nomialsrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 623456 21 pages 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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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

4 ISRN Discrete Mathematics

By (16) and (20) we obtain

119861119909119903=

infin

sum

119896=0

1

119896

119896

sum

119895=0

(minus1)119896minus119895

(

119896

119895) (119895 + 119903)

119909

=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(28)

Hence we have the following theorem

Theorem 10 The 119903-Bell numbers are equal to

119861119909 119903=

infin

sum

119896=0

119909 + 119903

119896 + 119903

119903

(29)

where 119909 and 119903 are complex numbers

By Theorem 10 we have verified that identity in (1) alsoholds for 119861

119909119903with complex argument 119909 This means that

the numbers 119861119909119903

in Definition 9 belong to the family of Bellnumbers

The next theorem asserts that 119861119909119903

also possess a kindof Dobinskirsquos formula which can easily be shown usingDefiniton 9 and the expansion of 119890(119903119911+119890

119911minus1)

Theorem 11 The 119903-Bell numbers are equal to

119861119909 119903=

1

119890

infin

sum

119896=0

(119896 + 119903)119909

119896

(30)

where 119909 and 119903 are complex numbers

Using (26) and (30) we obtain the following corollarywhich is a kind of extension of the integral formula obtainedby Mezo in [8]

Corollary 12 The following integral identity holds

infin

sum

119896=0

(119896 + 119903)119909

119896

=

119909119890

2120587119894

int

H

119890119903119911+(119890

119911minus1)119889119911

119911119909+1 (31)

The 119903-Bell polynomials of Mezo [8] satisfy the recurrencerelation

119861119909 119903(119908) = 119903119861

119909minus1 119903(119908) + 119908119861

119909minus1 119903+1(119908) (32)

where 119909 and 119903 are nonnegative integers This will reduce to

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

(33)

when119908 = 1 Analogous to this relation we have the followingrelation which can easily be shown using (28) (15) and (16)

Theorem 13 The 119903-Bell numbers satisfy the following relation

119861119909 119903= 119903119861119909minus1 119903

+ 119861119909minus1 119903+1

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(34)

where 119909 and 119903 are complex numbers

Proof Summing up both sides of (15) gives

119861119909 119903= (119903 + 1) 119861

119909minus1 119903+

119909 + 119903 minus 1

119903 minus 1

119903

+

infin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

(35)

By applying (16) we getinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

infin

sum

119896=0

[

[

1

(119896 minus 1)

119896

sum

119895=0

(minus1)119896minus1minus119895

(

119896 minus 1

119895) (120573119895 + 119903)

119909minus1]

]

minus

infin

sum

119896=0

[

[

minus1

(119896minus1)

119896

sum

119895=0

(minus1)119896minus1minus(119895minus1)

(

119896minus1

119895minus1) ((119895minus1)+1+119903)

119909minus1]

]

(36)

Hence we haveinfin

sum

119896=0

119896

119909 + 119903 minus 1

119896 + 119903

119903

= minus

119909 + 119903 minus 1

119903 minus 1

119903

minus 119861119909minus1 119903

+ 119861119909minus1 1+119903

+

119909 + 119903 minus 1

119903 minus 1

119903+1

(37)

Substituting this to (35) completes the proof of the theorem

Note that when the parameters 119909 119910 and 119903 are nonneg-ative integers Definition 1 is just equivalent to the integralrepresentation of the 119903-Stirling numbers of the second kindin [1] Hence the value of 119909+119903minus1

119903minus1119903+1

is equal to 0

4 Asymptotic Formula

An asymptotic formula for 119903-Bell numbers 119861119909119903

was firstestablished by Corcino in [7] But the formula holds onlywhen 119909 is a nonnegative integer Here we aim to obtain anasymptotic formula for 119861

119909119903when 119909 is a real number

Using Definition 9

119861119909119903=

119909

2120587119894

int

H

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (38)

To obtain an asymptotic formula we deform the pathH intothe following contour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 where

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number(ii) 1198622is the line segmentreal119911 = 120598 going from 120582+119894(120575minus2120587)

to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

ISRN Discrete Mathematics 5

The new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the counter-

clockwise senseThis idea of deforming the contourH is alsodone in [13] The integrals along 119862

1 1198622 1198624 and 119862

5are seen

to be

119874(

(2 + 120582)119910

(2120587 minus 120575)119909) (39)

It will also be shown that these integrals go to 0 as 119909 rarr infin

provided that 119909minus119910 ge 11990913 To see this we consider1198621 For the

other contours the estimate can be seen similarly We look at1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

le int

1198621

1003816100381610038161003816100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)

119911119909+1

1003816100381610038161003816100381610038161003816100381610038161003816

|119889119911| (40)

Note that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

=10038161003816100381610038161198901199031199111003816100381610038161003816

100381610038161003816100381610038161003816

119890(119890119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890real(119890119911minus1)

(41)

Turning to 119890119911 minus 1

119890119911

minus 1 = 119890real119911

119890119894image119911

minus 1

= [119890real119911 cosimage119911 minus 1] + 119894119890real119911 sinimage119911

(42)

Thus100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890119890real119911 cosimage119911

119890minus1

lt 119890119903real119911

119890119890real119911

119890minus1

(43)

Consider

119903real119911 + 119890real119911

minus 1 lt 119903120582 + 119890120582

minus 1

= 119903120582 + 120582(1 +

120582

2

+

1205822

3

+

1205823

4

+ sdot sdot sdot)

lt 119903120582 + 120582(1 +

120582

2

+

1205822

22+

1205823

23+ sdot sdot sdot)

= 119903120582 + 120582 (

2

2 minus 120582

)

(44)

Choose 120582 lt 1 so that 2 minus 120582 gt 1 Then

119903real119911 + 119890real119911

minus 1 lt 120582 (2 + 119903) (45)

This implies that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

lt 119890120582(2+119903)

(46)

Consequently we obtain1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

lt 119890120582(2+119903)

1

(2120587 minus 120575)119909+1119897 (1198621) (47)

where

119897 (1198621) = lim119905rarrminusinfin

120582 minus 119905 (48)

Since 119897(1198621)(2120587minus120575)

119909+1

rarr 0 as 119909 rarr infin we have the integralalong119862

1goes to 0 as 119909 rarr infin So what remains is the integral

along 1198623 That is

119861119909119903sim

119909

2120587119894

int

1198623

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (49)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Hence by

Laplace method or following the analysis in [7]

119861119909119903sim

119909

2120587119894119877119899int

120572

minus120572

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

sim

119909

2120587119894119877119899int

120598

minus120598

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

(50)

In [7] the integration is along a circle about zero with radius119877 This number 119877 is shown to be the unique solution to

120583

119909 minus 119903120583

= 119890minus120583 (51)

as a function of 120583 (see Lemma 3 [7]) We see that the asymp-totic formula for the 119903-Bell numbers obtained in [7] holdsfor real argument 119909 Thus we have the following asymptoticformula

Theorem 14 The 119903-Bell numbers 119861119909119903

with real arguments 119909and 119903 have the following asymptotic formula

119861119909119903sim 119862radic120587[1 +

119863 minus (21198774

+ 91198773

+ 161198772

+ 6119877 + 2)

24119890119877119877(119877 + 1 + 119903119890

minus119877)3

]

(52)

where

119862 =

119909 exp (119903119877 + 119890119877 minus 1)

120587119877119909[2 (1199032119890119877+ 119877119890119877+ 119903119877)]

12

119863 = (31198773

+ 81198772

minus 6119877 minus 4 minus 2119903119890minus119877

) 119903119890minus119877

(53)

and 119877 is the unique positive solution to

120583119890120583

+ 119903120583 minus 119909 = 0 (54)

as a function of 120583

An asymptotic formula for (119903 120573)-Bell numbers 119866119909119903120573

hasalready been established in [14] However the formula holdsonly when 119909 is a nonnegative integer Here using the samemethod as employed above we can show that this asymptoticformula will also work for the case in which the parameters119909 119903 and 120573 are real numbers

Now for real parameters 119909 119903 and 120573 we have

119866119909119903120573=

119909

2120587119894

int

H

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1 (55)

where the path H can also be deformed into the followingcontour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 such that

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number

6 ISRN Discrete Mathematics

(ii) 1198622is the line segmentreal119911 = 120582 going from 120582 + 119894(120575 minus

2120587) to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

Also the new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the

counterclockwise sense It can easily be shown that along thepath 119862

1

100381610038161003816100381610038161003816

119890119903119911+(119890

120573119911minus1)120573100381610038161003816100381610038161003816

lt 119890120582((2120573)+119903)

(56)

Hence as 119909 goes toinfin100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1

100381610038161003816100381610038161003816100381610038161003816

997888rarr 0 (57)

This can be done similarly along1198622 1198624 and119862

5Thus we have

119866119909119903120573sim

119909

2120587119894

int

1198623

119890119903119911+(119890

120573119911minus1)120573

119911119909+1

119889119911 (58)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Then

119866119909119903120573sim

119909

2120587119894119877

119899int

120572

minus120572

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

sim

119909

2120587119894119877

119899int

120598

minus120598

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

(59)

This implies that the asymptotic formula for the (119903 120573)-Bellnumbers obtained in [14] holds for real argumentThus from[14] we have the following asymptotic formula for (119903 120573)-Bellnumbers 119866

119909119903120573with real arguments 119909 119903 and 120573

119866119909119903120573

sim

11990912

(1 + 112119909) exp (119903119877 + 120573minus1119890120573119877 minus 120573 minus 119909) (120573120573119877 + 119903)119909

[(119909 minus 119903119877) 120573minus1]

12

(120573119877 + 1 + 119903119890minus120573119877)

12

times (1 +

119863 + 119864

119865

)

(60)

where

119863 = (31205732

119877

3

+ 8120573119877

3

+ 3120573119877 + 3 minus 10120573minus1

minus 2119903119890minus120573119877

) 119903119890minus120573119877

119864 = (31205733

minus 51205732

) 119877

4

+ (211205732

minus 30120573)119877

3

+ (39120573 minus 55) 119877

2

+ (24 minus 30120573minus1

) 119877 + (3120573minus1

minus 5120573minus2

)

119865 = 24119877119890120573119877

(120573119877 + 1 + 119903119890minus120573119877

)

3

(61)

and 119877 is the unique positive solution to

120583119890120573120583

+ 119903120583 minus 119909 = 0 (62)as a function of 120583

5 Summary and Recommendation

In this paper we have defined 119903-Stirling numbers of thesecond kind and 119903-Bell numbers with complex argumentsusing the concept of Hankel contour and established someproperties parallel to those of the classical Stirling and Bellnumbers Moreover we have derived an asymptotic formulafor 119903-Bell numbers as well as for (119903 120573)-Bell numbers for realarguments using the method of Chelluri and that of Moserand Wyman

We observe that by employing those methods one canpossibly establish an asymptotic formula for 119903-Stirling num-bers of the second kind with real arguments and conse-quently using relation (6) an asymptotic formula for (119903 120573)-Stirling numbers with real arguments

Acknowledgments

The authors would like to acknowledge the support from theOffice of the Vice Chancellor for Research and Developmentof the University of the Philippines Diliman for this researchproject The authors also wish to thank the referees forreading and evaluating the paper thoroughly

References

[1] A Z Broder ldquoThe r-Stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[2] I Mezo ldquoOn the maximum of r-Stirling numbersrdquo Advances inApplied Mathematics vol 41 no 3 pp 293ndash306 2008

[3] I Mezo ldquoNew properties of r-Stirling seriesrdquoActaMathematicaHungarica vol 119 no 4 pp 341ndash358 2008

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the 119903 120573-Stirling numbersrdquo Ars Combinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] C B Corcino ldquoAn asymptotic formula for the r-Bell numbersrdquoMatimyas Matematika vol 24 no 1 pp 9ndash18 2001

[8] I Mezo ldquoThe r-Bell numbersrdquo Journal of Integer Sequences vol14 no 1 article 1111 2011

[9] R B Corcino M B Montero and C B Corcino ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[10] L Moser and M Wyman ldquoAn asymptotic formula for the Bellnumbersrdquo vol 49 pp 49ndash54 1955

[11] R L Graham D E Knuth and O Patashnik Concrete Math-ematics Addison-Wesley Reading Mass USA 2nd edition1994

[12] P Flajolet and H Prodinger ldquoOn Stirling numbers for complexarguments and Hankel contoursrdquo SIAM Journal on DiscreteMathematics vol 12 no 2 pp 155ndash159 1999

[13] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

ISRN Discrete Mathematics 7

[14] R B Corcino and C B Corcino ldquoOn generalized Bell poly-nomialsrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 623456 21 pages 2011

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

ISRN Discrete Mathematics 5

The new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the counter-

clockwise senseThis idea of deforming the contourH is alsodone in [13] The integrals along 119862

1 1198622 1198624 and 119862

5are seen

to be

119874(

(2 + 120582)119910

(2120587 minus 120575)119909) (39)

It will also be shown that these integrals go to 0 as 119909 rarr infin

provided that 119909minus119910 ge 11990913 To see this we consider1198621 For the

other contours the estimate can be seen similarly We look at1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

le int

1198621

1003816100381610038161003816100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)

119911119909+1

1003816100381610038161003816100381610038161003816100381610038161003816

|119889119911| (40)

Note that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

=10038161003816100381610038161198901199031199111003816100381610038161003816

100381610038161003816100381610038161003816

119890(119890119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890real(119890119911minus1)

(41)

Turning to 119890119911 minus 1

119890119911

minus 1 = 119890real119911

119890119894image119911

minus 1

= [119890real119911 cosimage119911 minus 1] + 119894119890real119911 sinimage119911

(42)

Thus100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

= 119890119903real119911

119890119890real119911 cosimage119911

119890minus1

lt 119890119903real119911

119890119890real119911

119890minus1

(43)

Consider

119903real119911 + 119890real119911

minus 1 lt 119903120582 + 119890120582

minus 1

= 119903120582 + 120582(1 +

120582

2

+

1205822

3

+

1205823

4

+ sdot sdot sdot)

lt 119903120582 + 120582(1 +

120582

2

+

1205822

22+

1205823

23+ sdot sdot sdot)

= 119903120582 + 120582 (

2

2 minus 120582

)

(44)

Choose 120582 lt 1 so that 2 minus 120582 gt 1 Then

119903real119911 + 119890real119911

minus 1 lt 120582 (2 + 119903) (45)

This implies that100381610038161003816100381610038161003816

119890119903119911+(119890

119911minus1)100381610038161003816100381610038161003816

lt 119890120582(2+119903)

(46)

Consequently we obtain1003816100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911

1003816100381610038161003816100381610038161003816100381610038161003816

lt 119890120582(2+119903)

1

(2120587 minus 120575)119909+1119897 (1198621) (47)

where

119897 (1198621) = lim119905rarrminusinfin

120582 minus 119905 (48)

Since 119897(1198621)(2120587minus120575)

119909+1

rarr 0 as 119909 rarr infin we have the integralalong119862

1goes to 0 as 119909 rarr infin So what remains is the integral

along 1198623 That is

119861119909119903sim

119909

2120587119894

int

1198623

119890119903119911+(119890

119911minus1)

119911119909+1

119889119911 (49)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Hence by

Laplace method or following the analysis in [7]

119861119909119903sim

119909

2120587119894119877119899int

120572

minus120572

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

sim

119909

2120587119894119877119899int

120598

minus120598

exp (119890119877119890119894120579

+ 119903119877119890119894120579

minus 119894119899120579) 119889120579

(50)

In [7] the integration is along a circle about zero with radius119877 This number 119877 is shown to be the unique solution to

120583

119909 minus 119903120583

= 119890minus120583 (51)

as a function of 120583 (see Lemma 3 [7]) We see that the asymp-totic formula for the 119903-Bell numbers obtained in [7] holdsfor real argument 119909 Thus we have the following asymptoticformula

Theorem 14 The 119903-Bell numbers 119861119909119903

with real arguments 119909and 119903 have the following asymptotic formula

119861119909119903sim 119862radic120587[1 +

119863 minus (21198774

+ 91198773

+ 161198772

+ 6119877 + 2)

24119890119877119877(119877 + 1 + 119903119890

minus119877)3

]

(52)

where

119862 =

119909 exp (119903119877 + 119890119877 minus 1)

120587119877119909[2 (1199032119890119877+ 119877119890119877+ 119903119877)]

12

119863 = (31198773

+ 81198772

minus 6119877 minus 4 minus 2119903119890minus119877

) 119903119890minus119877

(53)

and 119877 is the unique positive solution to

120583119890120583

+ 119903120583 minus 119909 = 0 (54)

as a function of 120583

An asymptotic formula for (119903 120573)-Bell numbers 119866119909119903120573

hasalready been established in [14] However the formula holdsonly when 119909 is a nonnegative integer Here using the samemethod as employed above we can show that this asymptoticformula will also work for the case in which the parameters119909 119903 and 120573 are real numbers

Now for real parameters 119909 119903 and 120573 we have

119866119909119903120573=

119909

2120587119894

int

H

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1 (55)

where the path H can also be deformed into the followingcontour 119862

1cup 1198622cup 1198623cup 1198624cup 1198625 such that

(i) 1198621is the line image119911 = minus2120587 + 120575 120575 gt 0 and real119911 le 120582 120582 is a

small positive number

6 ISRN Discrete Mathematics

(ii) 1198622is the line segmentreal119911 = 120582 going from 120582 + 119894(120575 minus

2120587) to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

Also the new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the

counterclockwise sense It can easily be shown that along thepath 119862

1

100381610038161003816100381610038161003816

119890119903119911+(119890

120573119911minus1)120573100381610038161003816100381610038161003816

lt 119890120582((2120573)+119903)

(56)

Hence as 119909 goes toinfin100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1

100381610038161003816100381610038161003816100381610038161003816

997888rarr 0 (57)

This can be done similarly along1198622 1198624 and119862

5Thus we have

119866119909119903120573sim

119909

2120587119894

int

1198623

119890119903119911+(119890

120573119911minus1)120573

119911119909+1

119889119911 (58)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Then

119866119909119903120573sim

119909

2120587119894119877

119899int

120572

minus120572

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

sim

119909

2120587119894119877

119899int

120598

minus120598

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

(59)

This implies that the asymptotic formula for the (119903 120573)-Bellnumbers obtained in [14] holds for real argumentThus from[14] we have the following asymptotic formula for (119903 120573)-Bellnumbers 119866

119909119903120573with real arguments 119909 119903 and 120573

119866119909119903120573

sim

11990912

(1 + 112119909) exp (119903119877 + 120573minus1119890120573119877 minus 120573 minus 119909) (120573120573119877 + 119903)119909

[(119909 minus 119903119877) 120573minus1]

12

(120573119877 + 1 + 119903119890minus120573119877)

12

times (1 +

119863 + 119864

119865

)

(60)

where

119863 = (31205732

119877

3

+ 8120573119877

3

+ 3120573119877 + 3 minus 10120573minus1

minus 2119903119890minus120573119877

) 119903119890minus120573119877

119864 = (31205733

minus 51205732

) 119877

4

+ (211205732

minus 30120573)119877

3

+ (39120573 minus 55) 119877

2

+ (24 minus 30120573minus1

) 119877 + (3120573minus1

minus 5120573minus2

)

119865 = 24119877119890120573119877

(120573119877 + 1 + 119903119890minus120573119877

)

3

(61)

and 119877 is the unique positive solution to

120583119890120573120583

+ 119903120583 minus 119909 = 0 (62)as a function of 120583

5 Summary and Recommendation

In this paper we have defined 119903-Stirling numbers of thesecond kind and 119903-Bell numbers with complex argumentsusing the concept of Hankel contour and established someproperties parallel to those of the classical Stirling and Bellnumbers Moreover we have derived an asymptotic formulafor 119903-Bell numbers as well as for (119903 120573)-Bell numbers for realarguments using the method of Chelluri and that of Moserand Wyman

We observe that by employing those methods one canpossibly establish an asymptotic formula for 119903-Stirling num-bers of the second kind with real arguments and conse-quently using relation (6) an asymptotic formula for (119903 120573)-Stirling numbers with real arguments

Acknowledgments

The authors would like to acknowledge the support from theOffice of the Vice Chancellor for Research and Developmentof the University of the Philippines Diliman for this researchproject The authors also wish to thank the referees forreading and evaluating the paper thoroughly

References

[1] A Z Broder ldquoThe r-Stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[2] I Mezo ldquoOn the maximum of r-Stirling numbersrdquo Advances inApplied Mathematics vol 41 no 3 pp 293ndash306 2008

[3] I Mezo ldquoNew properties of r-Stirling seriesrdquoActaMathematicaHungarica vol 119 no 4 pp 341ndash358 2008

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the 119903 120573-Stirling numbersrdquo Ars Combinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] C B Corcino ldquoAn asymptotic formula for the r-Bell numbersrdquoMatimyas Matematika vol 24 no 1 pp 9ndash18 2001

[8] I Mezo ldquoThe r-Bell numbersrdquo Journal of Integer Sequences vol14 no 1 article 1111 2011

[9] R B Corcino M B Montero and C B Corcino ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[10] L Moser and M Wyman ldquoAn asymptotic formula for the Bellnumbersrdquo vol 49 pp 49ndash54 1955

[11] R L Graham D E Knuth and O Patashnik Concrete Math-ematics Addison-Wesley Reading Mass USA 2nd edition1994

[12] P Flajolet and H Prodinger ldquoOn Stirling numbers for complexarguments and Hankel contoursrdquo SIAM Journal on DiscreteMathematics vol 12 no 2 pp 155ndash159 1999

[13] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

ISRN Discrete Mathematics 7

[14] R B Corcino and C B Corcino ldquoOn generalized Bell poly-nomialsrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 623456 21 pages 2011

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

6 ISRN Discrete Mathematics

(ii) 1198622is the line segmentreal119911 = 120582 going from 120582 + 119894(120575 minus

2120587) to the circle |119911| = 119877(iii) 119862

5and119862

4are the reflections in the real axis of 119862

1and

1198622 respectively

(iv) 1198623is the portion of the circle |119911| = 119877 meeting 119862

2and

1198624

Also the new contour is 1198621cup 1198622cup 1198623cup 1198624cup 1198625in the

counterclockwise sense It can easily be shown that along thepath 119862

1

100381610038161003816100381610038161003816

119890119903119911+(119890

120573119911minus1)120573100381610038161003816100381610038161003816

lt 119890120582((2120573)+119903)

(56)

Hence as 119909 goes toinfin100381610038161003816100381610038161003816100381610038161003816

int

1198621

119890119903119911+(119890

120573119911minus1)120573

119889119911

119911119909+1

100381610038161003816100381610038161003816100381610038161003816

997888rarr 0 (57)

This can be done similarly along1198622 1198624 and119862

5Thus we have

119866119909119903120573sim

119909

2120587119894

int

1198623

119890119903119911+(119890

120573119911minus1)120573

119911119909+1

119889119911 (58)

where 1198623is a semicircle 119911 = 119877119890119894120579 minus120572 le 120579 le 120572 Then

119866119909119903120573sim

119909

2120587119894119877

119899int

120572

minus120572

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

sim

119909

2120587119894119877

119899int

120598

minus120598

exp (120573minus1119890120573119877119890119894120579

+ 119903119877119890119894120579

minus 119894119909120579 minus 120573minus1

) 119889120579

(59)

This implies that the asymptotic formula for the (119903 120573)-Bellnumbers obtained in [14] holds for real argumentThus from[14] we have the following asymptotic formula for (119903 120573)-Bellnumbers 119866

119909119903120573with real arguments 119909 119903 and 120573

119866119909119903120573

sim

11990912

(1 + 112119909) exp (119903119877 + 120573minus1119890120573119877 minus 120573 minus 119909) (120573120573119877 + 119903)119909

[(119909 minus 119903119877) 120573minus1]

12

(120573119877 + 1 + 119903119890minus120573119877)

12

times (1 +

119863 + 119864

119865

)

(60)

where

119863 = (31205732

119877

3

+ 8120573119877

3

+ 3120573119877 + 3 minus 10120573minus1

minus 2119903119890minus120573119877

) 119903119890minus120573119877

119864 = (31205733

minus 51205732

) 119877

4

+ (211205732

minus 30120573)119877

3

+ (39120573 minus 55) 119877

2

+ (24 minus 30120573minus1

) 119877 + (3120573minus1

minus 5120573minus2

)

119865 = 24119877119890120573119877

(120573119877 + 1 + 119903119890minus120573119877

)

3

(61)

and 119877 is the unique positive solution to

120583119890120573120583

+ 119903120583 minus 119909 = 0 (62)as a function of 120583

5 Summary and Recommendation

In this paper we have defined 119903-Stirling numbers of thesecond kind and 119903-Bell numbers with complex argumentsusing the concept of Hankel contour and established someproperties parallel to those of the classical Stirling and Bellnumbers Moreover we have derived an asymptotic formulafor 119903-Bell numbers as well as for (119903 120573)-Bell numbers for realarguments using the method of Chelluri and that of Moserand Wyman

We observe that by employing those methods one canpossibly establish an asymptotic formula for 119903-Stirling num-bers of the second kind with real arguments and conse-quently using relation (6) an asymptotic formula for (119903 120573)-Stirling numbers with real arguments

Acknowledgments

The authors would like to acknowledge the support from theOffice of the Vice Chancellor for Research and Developmentof the University of the Philippines Diliman for this researchproject The authors also wish to thank the referees forreading and evaluating the paper thoroughly

References

[1] A Z Broder ldquoThe r-Stirling numbersrdquo Discrete Mathematicsvol 49 no 3 pp 241ndash259 1984

[2] I Mezo ldquoOn the maximum of r-Stirling numbersrdquo Advances inApplied Mathematics vol 41 no 3 pp 293ndash306 2008

[3] I Mezo ldquoNew properties of r-Stirling seriesrdquoActaMathematicaHungarica vol 119 no 4 pp 341ndash358 2008

[4] R B Corcino C B Corcino and R Aldema ldquoAsymptoticnormality of the 119903 120573-Stirling numbersrdquo Ars Combinatoria vol81 pp 81ndash96 2006

[5] I Mezo ldquoA new formula for the Bernoulli polynomialsrdquo Resultsin Mathematics vol 58 no 3-4 pp 329ndash335 2010

[6] A Rucinski and B Voigt ldquoA local limit theorem for generalizedStirling numbersrdquo Revue Roumaine de Mathematiques Pures etAppliquees vol 35 no 2 pp 161ndash172 1990

[7] C B Corcino ldquoAn asymptotic formula for the r-Bell numbersrdquoMatimyas Matematika vol 24 no 1 pp 9ndash18 2001

[8] I Mezo ldquoThe r-Bell numbersrdquo Journal of Integer Sequences vol14 no 1 article 1111 2011

[9] R B Corcino M B Montero and C B Corcino ldquoOn general-izedBell numbers for complex argumentrdquoUtilitasMathematicavol 88 pp 267ndash279 2012

[10] L Moser and M Wyman ldquoAn asymptotic formula for the Bellnumbersrdquo vol 49 pp 49ndash54 1955

[11] R L Graham D E Knuth and O Patashnik Concrete Math-ematics Addison-Wesley Reading Mass USA 2nd edition1994

[12] P Flajolet and H Prodinger ldquoOn Stirling numbers for complexarguments and Hankel contoursrdquo SIAM Journal on DiscreteMathematics vol 12 no 2 pp 155ndash159 1999

[13] R Chelluri L B Richmond and N M Temme ldquoAsymptoticestimates for generalized Stirling numbersrdquoAnalysis vol 20 no1 pp 1ndash13 2000

ISRN Discrete Mathematics 7

[14] R B Corcino and C B Corcino ldquoOn generalized Bell poly-nomialsrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 623456 21 pages 2011

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

ISRN Discrete Mathematics 7

[14] R B Corcino and C B Corcino ldquoOn generalized Bell poly-nomialsrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 623456 21 pages 2011

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