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J~n –an –abha ßbDbBßbDbBßbDbBßbDbBßbDbB

VOLUME 41

2011

ISSN 0304-9892

Published by :

The Vij~n–ana Parishad of IndiaDAYANAND VEDIC POSTGRADUATE COLLEGE

(Bundelkhand University)ORAI, U.P., INDIA

www.vijnanaparishadofindia.org

J~N – AN – ABHAEDITORS

H. M. Srivastava R.C. Singh ChandelUniversity of Victoria AND D.V. Postgraduate CollegeVictoria, B.C., Canada Orai, U.P., India

Editorial Advisory BoardR.G. Buschman (Langlois, OR) K.L. Chung (Stanford)A. Carbone (Rende, Italy) N.E. Cho (Pusan, Korea)R.C. Chaudhary (Jaipur) Sunil Datta (Lucknow)R.K. Datta (Delhi) Madhu Jain (Roorkee)P.W. Karlsson (Lyngby, Denmark) Karmeshu (Delhi)I. Massabo (Rende, Italy) Pranesh Kumar (Prince George, BC, Canada)G.V. Milovanovie (NIS, Serbia) S. Owa (Osaka, Japan)B.E. Rhoades (Bloomington, IN) D. Roux (Milano, Italy)T.M. Rassias (Athens, Greece) L.J. Slater (Cambridge, U.K.)V.P. Saxena (Bhopal) K.N. Srivastava (Bhopal)Dinesh Singh (Delhi) S.P. Singh (London, ON, Canada)M.R. Singh (London, ON, Canada) R.P. Singh (Petersburg)J.N. Singh (Miami Shores, Florida) T. Singh (BITS, Pilani, Goa Campus)R.K. Tuteja (Rohtak)

Vij~n –ana Parishad of India(Society for Applications of Mathematics)

(Registered under the Societies Registration Act XXI of 1860)Office : D.V. Postgraduate College, Orai-285001, U.P., India

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: Madhu Jain (Roorkee): Principal

D.V. Postgraduate College, (Orai)Secretary-Treasurer : R. C. Singh Chandel (Orai)Foreign Secretary : H.M. Srivastava (Victoria)

MEMBERSG.C. Sharma (IPP) (Agra) R.D. Agrawal (Vidisha)V.P. Saxena (Bhopal) Karmeshu (Delhi)P. Chaturani (Mumbai) Renu Jain (Gwalior)A.P. Dwivedi (Keetham, Agra) D.S. Hooda (Raghogarh)M.N. Mehta (Surat) B.S. Bhadauria (Lucknow)K.R. Pardasani (Bhopal) S.S. Chauhan (Orai)H. Kumar (Kanpur) S.S. Agrawal (Meerut)

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CONTENTCERTAIN FRACTIONAL CALCULUS OPERATORS ASSOCIATED WITH FOX-WRIGHT GENERALIZED HYPERGEOMETRIC FUNCTION

-Renu Jain and Dinesh SinghLIE THEORY AND BASIC GAUSS POLYNOMIALS

-Kishan Sharma and Renu JainRECURRENCE RELATIONS FOR THE H -FUNCTION

-S.C. Sharma and Rachna BhargavaCONVEX OPTIMIZATION TECHNIQUES DUE TO NESTROV ANDCOMPUTATIONAL COMPLEXITY

- Paras Bhatnagar and Prashant ChauhanON DECOMPOSITION OF CURVATURE TENSOR FIELDS IN A KAEHLERIANRECURRENT SPACE OF FIRST ORDER

-K.S. Rawat and Nitin UniyalFOURIER SERIES FOR MODIFIED MULTI-VARIABLE H-FUNCTION

-Rashmi Singh and R.S. ChandelA REMARK ON "FRACTIONAL CALCULUS OPERATOR INVOLVING THEPRODUCT OF HYPERGEOMETRIC FUNCTION OF SEVERAL VARIABLES" BYV.B.L. CHAURASIA AND HARI SINGH PARIHA J~n–an–abha, 35, (2005), 175-187

-R.C. Singh ChandelPHASE SHIFTS OF S-WAVE SCHRÖDINGER EQUATION FOR MITTAG- LEFFLERFUNCTION

- Hemant Kumar and Vimal Pratap SinghA SHORT NOTE ON EXTON'S RESULT

- M. Kamarujjama and Waseem A. KhanAPPLICATIONS OF SAIGO FRACTIONAL CALCULUS OPERATORS FORCERTAIN SUBCLASS OF MULTIVALENT FUNCTIONS WITH NEGATIVECOEFFICIENTS

- B.B. Jaimini and Jyoti GuptaSOME RESULTS ON A -GENERALIZED RIEMANN ZETA FUNCTION

- R.K. Gupta and Mina KumariSOLUTION OF QUADRATIC DIOPHANTINE EQUATIONS

-Pratap Singhq-OPERATIONAL FORMULAE FOR A CLASS OF q-POLYNOMIALS UNIFYINGTHE GENERALIZED q-HERMITE AND q-LAGUERRE POLYNOMIALS

- R.K. Kumbhat, R.K. Gupta and Madhu ChauhanON DISCRETE HERMITE TRANSFORM OF GENERALIZED FUNCTIONS

- S.R. MahtoDRS–TRIANGULATION BASED ON 2N–RAY ALGORITHM

- R.S. Patel, Ankit Agarwal and Lakshmi Narayan TripathiCOMPARISON OF A.D.D. OF TRIANGULATIONS

- R.S. Patel, Ankit Agarwal and Preeti Singh BhahelON SOME GENERALIZED FRACTIONAL DERIVATIVE FORMULAS

-R.C. Singh Chandel and Vandana GuptaSTRONG CONVERGENCE THEOREMS FOR UNIFORMLY EQUI-CONTINUOUSAND ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS

-Gurucharan Singh SalujaIMPACT OF SEXUAL MATURATION ON THE TRANSMISSION DYNAMICS OFHIV INFECTION IN HETEROGENEOUS COMMUNITY : A MODEL AND ITSQUALITATIVE ANALYSIS

-Abha Teguria, Manindra Kumar Srivastava and Anil RajputON UNIT OF LENGTH MEASUREMENT IN THE INDUS-SARASWATICIVILIZATION AND SPEED OF LIGHT IN THE VEDIC LITERATURE

-M.R. Goyal

...1-8

...9-11

...13-16

...17-26

...27-32

...33-39

...40

...41-46

...47-50

...51-62

...63-68

...69-71

...73-80

...81-84

...85-94

...95-108

...109-130

...131-140

...141-154

...145-159

J~n–an–abha, Vol. 41, 2011

CERTAIN FRACTIONAL CALCULUS OPERATORS ASSOCIATED WITHFOX-WRIGHT GENERALIZED HYPERGEOMETRIC FUNCTION

ByRenu Jain

School of Mathematics and Allied SciencesJiwaji University, Gwalior-474009, Madhya Pradesh, India

E-mail : [email protected]

Dinesh Singh

Department of Applied Mathematics, ITM Gwalior-474001Madhya Pradesh, India

E-mail : [email protected](Received : December 20, 2009; Revised : May 15, 2010, Final Form : October 10, 2011)

ABSTRACTIn this paper Riemann-Lioville fractional integral and derivative formulas

for Fox-Wright p q z generalized hypergeometric functions are obtained. Somespecial cases of the established formulas are also discussed.2000 Mathematics Subject Classification : 26A33, 45D05.Keywords : Fox-Wright generalized hypergeometric Function, Riemann-Liuvillefractional Integral and differential.

1. Introduction. The Fox-Wright generalized hypergeometric p q z for

z C is defined in series form as [4].

1 1

01

1

pk

i ji i ,p ip q p q q

ki j ,q i ji

a A k za ,Az z

b ,B b B k k!...(1.1)

where 0 0j j i ja ,b C,A ,B

1 1

1 0 0 1 1q p

j i i j i jj j

B A ; A ,B R A ,B i ,..., p, j ,...,q for suitably

bounded value of z .

As special case, for A1=...=Ap=1, B1=...=Bq=1, (1.1) reduces to generalizedhypergeometric function [3]

1 1 1

111

1 1

1 1

p p pp q p q

qqq

a , ,..., a , a ,..., a a ,...,a ;z F z

b ,...,b ;b ,..., bb , ,..., b ,

...(1.2)

2

For p=1, q=1, 1 1 11a , A ,b and 1B , (1.1) reduces to

1 10

1 k

,k

, k zz E z

, k k! ...(1.3)

called generalized Mittag-Leffler function defined by Prabhakar [2].

If 1 then (1.3) reduces to generalized Mittag-Leffler function

1 1

0

1 1 1 k

,k

, k zz E z

, k k! .

If 1 and 1 then (1.3) reduces to

1 1

0

1 1 11 1

k

k

, k zz E z

, k k! ...(1.3)

called Mittag-Leffler function [1].The object of this paper is to derive the relations which exist between the

Fox-Wright generalized hypergeometric function p q z and left and right side

Riemann-Liouville fractional integral and differential operators.The fractional integral and differential operators defined by Samko, Kilbas

and Marichev [7] for >0 are given by

1

0 0

1 xI f x x t f t dt , ...(1.4)

11a

xI f x t x f t dt , ...(1.5)

1

0 0

11

xdD f x x t f t dt

dx , ...(1.6)

11

1 x

dD f x t x f t dt

dx , ...(1.7)

where [ ] means the maximal integer not exceeding and {} is the fractionalpart of .

2.Properties of generalized Fox-Wright hypergeometric function.In this section we derive several interesting properties of the generalized Fox-

3

Wright hypergeometric function p q z defined by (1.1) with the help of Riemann

-Liouville fractional integral and derivative formula [4].

Theorem 1. Let 0 0 0, , and a R , let 0aI be left sided operator of

Riemann-Liouville fractional integral (1.4). Then there holds the formula

1 11 10 1 1

11 1

i i i i,p ,p

p q p qj j j,q ,q

a , A a , A , ,I t at x x ax

b ,B b ,B , , ...(2.1)

Proof.By virtue of (1.1) and (1.4) we have

1 111 1

0 00

1

1

1

p

k ki ii i x,pa ip q q

kj j ,q j jj

a A ka ,A x t a tI t at x dt .

k!b ,B b B k

Interchanging the order of integration and summation; and evaluating the innerintegral with the help of beta functions, by setting t=xy, we get

1 1 1 11

00

1

1

p

k ki iki

qk

i jj

a A ka x

L.H.S. y y dyk!b B k

111 1

1

i i ,p

p qi j ,q

a ,A , ,x ax .

b ,B , ,

Interchanging the order of integration and summation is permissible under theconditions stated with the theorem, due to convergence of the integral involved inthe process. This completes the proof of Theorem 1.

Corollary 1.1 .For , , , , 0 0 0 there holds the formula

,

,I t at x x E ax

,

1 10 1 1

1...(2.2)

Corollary 1.2 . By setting 1 in (2.2) there holds the formula

1 1

0 , ,I t E at x x E ax ...(2.3)

establised by Saxena and Saigo [6].

Theorem 2. Let , , 0 0 0 and a R , let I be right operator of Riemann

Liouville fractional integral (1.5). Then there holds the formula

4

i i i i,p ,p

p q p qj j j j,q ,q

a ,A a ,A , ,I t at x x ax

b ,B b ,B , ,

1 1

1 1

1 1

...(2.4)

Proof. By virtue of (1.1) and (1.5), we have

11

1

01

1

1

p

k ki ii i ,pi

p q qxkj j ,q j j

j

a A ka ,A t x a tI t at x dt

k!b ,B b B k.

Interchanging the order of integration and summation and then evaluating theinner integral by beta function formula, we get

1 11

0

1

p

ki iki

q xk

i ij

a A ka

L.H.S. t x t dtk!b B k

11

00

1

p

ki iki

qk

j jj

a A ka

y x y dtk!b B k

(by putting t–x=y)

i i ,p

p qj j ,q

a ,A , ,x ax

b ,B , ,

1

1 1

1

(putting y=wx).

Interchanging the order of integration and summation is permissible under theconditions stated with the theorem, due to convergence of the integrals involvedin the process. This completes the proof of Theorem 2.

Corollary 2.1. For , , ,a R 0 0 0 , there holds the formula

,

,I t at x x E ax

,

1 1

1. ...(2.5)

Corollary 2.2. By setting 1 in (2.5), there holds the formula

,

,I t at x x E ax

,

1 1

1 1...(2.6)

5

estabilished by Saxena and Saigo [6].

Theorem 3. Let , , , 0 0 0 , a R and D0 be left sided operator of

Riemann Liouville fractional integral (1.6). Then there holds the formula

i i i i,p ,p

p q p qj j j j,q ,q

a ,A a ,A , ,D t t at x x ax

b ,B b ,B , ,

1 11 1 10 1 1 1 1

1 1

...(2.7)


11 111 1

0 0

1 1

i i i i,p ,p

p q p qj j j j,q ,q

a ,A a ,AdD t at x I t at x

dxb ,B b ,B

1

1 1 11

00

1

1

p

ki i x k yiq

kj j

j

a A ka d

x t t dt.dxk!b B k

Interchanging the order of integration and summation and evaluating theinner integral by beta function formula (by setting t=xy), we obtain

11

0

1

1

p

i i kki

qk

j jj

a A kka d

L.H.S. xdx

k! kb B k

p

i i k kiq

kj j

j

a A kk x a

k k!b B k

11

0

1

i i ,p

p qj j ,q

a ,A , ,x ax

b ,B , ,

111 1

1

.

Interchaning the order of integration and summation is permissible under thecondition stated with the theorem, due to convergence of the integral involved inthe process. This completes the proof of Theorem 3.

6

Corollary 3.1. For , , , , 0 0 0 there holds the formula

,

,D t at x x E ax

,

1 10 1 1

1...(2.8)

Corollary 3.2. By setting 1 in (2.8) then there holds the formula

,

,D t at x x E ax

,

1 10 1 1

1 1...(2.9)

established by Saxena and Saigo [5].

Theorem 4. Let , , , 0 0 0 , a R and D be right sided operator of

Riemann Liouville fractional integral (1.7). Then there holds the formula

1 1

1 1

1 1

i i i i,p ,p

p q p qj j j j,q ,q

a , A a ,A , ,D t at x x ax

b ,B b ,B , , ...(2.10)


11 11

1 1

i i i i,p ,p–p q – p q

j j j j,q ,q

a ,A a ,AdD t at x I t at x

dxb ,B b ,B

11

0

1

1

p

i i kk yi

q xk

j jj

a A ka d

t x t dtdxk!b B k

11 11

00

1

1

p

i i kki

qk

j jj

a A ka d

x y y dydxk!b B k

(putting t–x=y)

111

0

1

1

p

ki i– k–i

qk

j jj

a A k a k dx

k! k dxb B k (putting y=wx )

7

p

ki iki

qk

j jj

a A ka k

xk! k

b B k

1

0

1

1

1 1

1

i i ,p –p q

j j ,q

a ,A , ,x ax .

b ,B , ,

Interchanging the order of integration and summation is permissible under thecondition stated with the Theorem, due to convergence of the integral involved inthe process. This completes the proof of Theorem 4.

Corollary 4.1. For , , , 0 0 0 , there holds the formula

1 1

1, –

,D t at x x E ax

, ...(2.11)

Corollary 4.2 . For 1 , (2.11) takes the form

1 1

1 1, –

,D t at x x E ax

, ...(2.12)

established by Saxena and Saigo [6].Acknowledgements

The authors are thankful to the referee for his valuable suggestions tobring the paper in its present form.

REFERENCES[1] A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized Mittag-Leffler function and

generalized fractional calculus operators, Integral Transform. Spec. Funct., 15 (2004),31-49.

[2] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler functionin the kernel, Yokohama Math. J., 19 (1971), 7-15.

[3] E.D. Rainville, Special Functions, Macmillan Co., New York (1960); Reprinted : ChelseaPubl. Co. Bronx, New York 1971.

[4] J. Ram and S. Chandak, Unified fractional derivative formula for the Fox-Wrightgeneralized hypergeometric function, Proc. Nat. Acad. Sci., India Sect A, 79 (2009), 51-57.

[5] H.M. Srivastava, Some Fox-Wright generalized hypergeometric function and associatedfamilies of convolution operators; Applicable Analysis and Discrete Methematics, 1 (2007),56-71.

[6] R.K. Saxena and M. Saigo, Certain properties of fractional operators associated with

8

generalized Mittag-Leffler function, Fractional Calculus and Applied Analysis, 8 (2005),141-154.

[7] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivative Theoryand Applications, Gordon and Breach New York (1993).

9


LIE THEORY AND BASIC GAUSS POLYNOMIALSBy

Kishan Sharma

Department of Applied MathematicsNRI Institute of Technology and Management, Gwalior-474001

Madhya Pradesh, IndiaE-Mail:[email protected], [email protected]

andRenu Jain

School of Mathematics and Allied SciencesJiwaji University, Gwalior-474009, Madhya Pradesh, India


(Received : May 15, 2010, Revised: October 15, 2011)

ABSTRACTIn the Present paper, an attempt has been made to bring basic

hypergeometric functions within the purview of Lie theory by constructing adynamical symmetry algebra of basic hypergeometric function 21. Multiplierrepresentation theory is then used to obtain generating function for basic analogueof Gauss polynomial. The results obtained in this paper are extensions of theresults derived earlier by Miller [3] and Sarkar-Chatterjea [4].2000 Mathematics Subject Classification : Primary 60E07, 20G05; Secondary33C55, 14A17.Keywords : Lie algebra, Generating function, Basic Gauss polynomials.

1. Introduction. The q-analogue of the Gauss functions or Heine's series[1] may be written as

n

a,b;c;q; x a;q,n b;q,n / c;q,n n;q !

2 10

c , , ,... 0 1 2

where q 1 and x 1 .

Here [a;q,n] and [n;q]! are respectively the basic Pochammer's symbol and basicfactorial function defined as [a;q,n]=[a;q][a+l;q]...[a+n–1;q] and

n;q ! ;q ;q ... n;q . 1 2

The basic differential operator q ,xB is defined by [1] through the relation

q,xB x qx x / x q 1 .... (1.1)

2. The Dynamical Symmetry Algebra of 2 1 . The dynamical symmetry

10

algebra of the hypergeometric function has been defined by Miller [2]. We use thesame technique to define the dynamical symmetry algebra of 2 1 . Let

, , ,q q q q/ . , ; ;q; x s u t 2 1 ...(2.1)

be the basis elements of a subspace of analytical functions of four variables x,s,uand t, associated with Heine's basic hypergeometric function of Heine's series 2 1 .Introduction of variables s,u and t renders differential operators independent ofparameters , and and thus facilitates their repeated operation.

The dynamical symmetry algebra of 2 1 is a 15-dimensional complex Lie

algebra isomorphic to sl (4), generated by twelve E -operators termed as raisingor lowering operators in view of their effect of raising or lowering the correspond-

ing suffix in ,q . The E -operators are

(i) ,q q,x q,t q ,s q ,uE s x x B tB sB xuB , 1 1

(ii) , ,q q ,x q,x q,tE u t x x B xsB tB , 1 1 1 1 ...(2.2)

The action of these operators on ,q is given by

,q , , ,q ,q , , ,qE ;q 11

, ,q , , ,q , q q ,qE ;q 1 11 ...(2.3)

The upper factor in each bracket is to be associated with plus sign andlower with minus sign. Twelve E-operators together with three maintenance

operators J ,J ,J and Identity operator I form a basic for gl(4) sl(4)(I), where(I) is the 1-dimensional Lie algebra generated by 1.Here

,q q ,s ,q q,u ,q q ,tJ sB J uB ,J tB and I 1 ...(2.4)

with the results

,q , , ,q , , ,qJ ;q ,

,q , , ,q , , ,qJ ;q ,

,q , , ,q , , ,qJ ;q ,

and

, , ,q , , ,qI . ...(2.5)

3. The Generating Functions for Basic Analogues of GaussPolynomials. On comparing the results obtained by the action one parameter

subgroup q , ,qexp aE generated by the operator ,qE

defined in (2.2) on , , ,q

defined in (2.1) and direct expansion, we get the identity

11

1

2 11st / ax st a / st , ; ;q; x a st / ax st

mm m mm

m

a m;q / m;q ! q , ; ;q ;q; x st

2 10

...(3.1)

Taking, m, m , q ,st ,a , 0 1 1 1 we get

112 1

0

1 2 1m

m

x ;q ;q / m;q ! . m, m ; ;q; x ...(3.2)

By definition of basic Gauss polynomial [1]

,mG q; x m, m ; ;q; x 2 1 1 ,

where , , , ,... 0 1 2 3 . ...(3.3)Using (3.3) in (3.2), we get the generating function

11

0

1 2 ,mm

m

x ;q ;q / m;q ! .G q; x

for basic Gauss polynomials.Acknowledgement

The authors are very much thankful to the referee for giving useful sug-gestions in the improvement of the paper.

REFERENCES[1] H. Exton, q-Hypergeometric Functions and Applications. Ellis Horwood Limited, John

Wiley and Sons, New York, Brisbane, Chichester and Toronto (1983).[2] W.Jr. Miller, Lie Theory and Special Functions, Academic Press. New York (1968).[3] W. Jr. Miller : Lie Theory and Generalization of Hypergeometric Functions, SIAM. J.

Appl. Math., 25 No. 2 (1973).[4] S. Sarkar and S.K. Chatterjea, Extensions of a class of bilateral generating relations for

certain special functions by group theoretic method, Pure Math. Manuscript, 9 (1990-91), 251-259.

13


RECURRENCE RELATIONS FOR THE H -FUNCTIONBy

S.C. Sharma

Department of Mathematics, University of RajasthanJaipur-302055, Rajathan, India

E-mail:[email protected]

Rachna Bhargava

Vivekanand College of Engineering (VIT Campus)Jaipur-302012, Rajasthan, India


(Received : May 16, 2010)

ABSTRACTThe aim of the present paper is to establish two new recurrence relations

for the H -function. Some results for the generalized Wright hypergeometricfunction and generalized Riemann Zeta function are special cases of our mainfindings.2000 Mathematics Subject Classification : 33C60Keywords and Phrases : Generalized Wright Hypergeometric Function,

Generalized Riemann Zeta Function and H -function.

1. Introduction. The H -function was introduced by Inayat Hussain [4]

and studied by Bushman and Srivastava [1].

The H -function is defined and represented in the following manner:

j j j j j,N N ,PM ,N M ,NP ,Q P ,Q

j j j j j,M M ,Q

a , ; A , a ,H z H z

b , , b , ,B

1 1

1 1

L

z d , z 1

02

...(1.1)

where

j

j

M N A

j j j jj j

Q PB

j j j jj M j N

b a

b a

1 1

1 1

1

1...(1.2)

14

The behaviour of H -function for small valkues of |z| follows easily from a result

recently given by Rathie {[6],p.306, eq.(6.9)}.2. Main Results. In this section, we establish two recurrence relations.

First Recurrence Relation


j j j j j,M M ,Q

a , ; A , a ,H z H z

b , , b , ,B

1 1

1 1

M Mj j j j j,N N ,Pb M.N

P ,Q

j j j j j,M M ,Q

a , ; A , a ,e H ze

b , , b , ,B

1 11 1

1 1

12

M Mj j j j j,N N ,Pb M ,N

P ,Q

j j j j j,M M ,Q

a , ; A , a ,e H ze

b , , b , ,B

1 11 1

1 1 2

...(2.1)

Second Recurrence Relation


j j j j j,M M ,Q

a , ; A , a ,H z H z

b , , b , ,B

1 1

1 1

N Nj j j j j,N N ,Pa aM.N

P ,Q

j j j j j,M M ,Q

a , ; A , a ,e H ze

b , , b , ,B

1 11 2

1 1

12

N Nj j j j j,N N ,Pa aM ,N

P ,Q

j j j j j,M M ,Q

a , ; A , a ,e H ze

b , , b , ,B

1 11 1 2

1 1

...(2.2)

Recurrence relations (2.1) and (2.2) are valid under the conditions of (1.1) and(1.2).Proof. The recurrence relations (2.1) and (2.2) can be easily established by appeal-ing to the definition [4], [1] and use the well known relation (Rainville [5], p.21,Th.8)

z zz zsin z e e

2

1 ,

involving -function and reinterpret the result in terms of the H -function, we

arrive at the right hand side of the desire results (2.1) and (2.2).

15

3. Particulars Cases.

(i) If in the recurrence relation (2.1) we reduce M ,NP ,QH to generalized Wright

hypergeometric function [3, p.271, eq. (7)]

1

11 1

01

1

12

j

M

M

j

P At

j jj j j ,P jbP Q P B

tj j j ,Q j jj M

a ta ; ; A ; zez e

t !b ; ; B ; b t

j

M

M

j

P At

j jjbP B

tj j

j M

a tze

et !b

1

1 1

0

1

.

(ii) If in the recurrence relation (2.2), we reduce M ,NP ,QH to generalized Riemann

Zeta function [3,p.27, eq.(1); 7, p.314-315, eq. (1.6) and (1.7)]

N Na a,,

, , , , ; Pz, p, e H ze

, , , ; P

1 11 22 2

0 1 1 1 110 1 12

N Na a,,

, , , , ;Pe H ze

, , , ; P

1 11 22 2

0 1 1 1 10 1 1 .

(iii) If Aj=Bj=1 in (2.1) and (2.2) then H reduces to Fox H-function and we getknown recurrence relations given earlier by Chaurasia [2].

ACKNOWLEDGEMENTThe authors are thankful to Professor H.M. Srivastava of the University

of Victoria, Canada for his kind help and suggestions in the preparation of thispaper.

REFERENCES[1] R.G. Bushman and H.M. Srivastava, The H-function associated with a certain class of

Feynman integrals, J. Phys. A: Math. Gen., 23 (1990), 4707-4710[2] V.B.L. Chaurasia, Investigation in Integral Transform and Special Functions, Ph.D.

Thesis, Univ. of Raj., Jaipur (India) 1976.[3] K.C. Gupta, R. Jain and A. Sharma, A study of unified finite integral transform with

Applications, J. Rajasthan Acad. Phy. Sci. 2(4) (2003), 269-282.[4] A.A. Inayat-Hussain, New Properties of Hypergeometric Series Derivable from Feynman

Integrals: II A Generalization of the H-function, J. Phys. A. Math. Gen. 20 (1987).[5] E.D., Rainville, Special Functions, Macmillan, New York, (1960); Reprinted : Chelsea

Publication Co. Boronx., New York 1971.

16

[6] A.K., Rathie, Le Mathematiche Fasc II,52 (1997), 297-310.[7] H.M. Shrivastava, K.C. Gupta and S.P. Goyal, The H-function of One and Two Variables

with Applications, South Asian Publishers, New Delhi and Madras (1982).

17


CONVEX OPTIMIZATION TECHNIQUES DUE TO NESTROV ANDCOMPUTATIONAL COMPLEXITY

ByParas Bhatnagar and Prashant Chauhan

Department of Mathematics, College of Engineering and Technology (IETM)Moradabad, Uttar Pradesh, India


ABSTRACTA large body of literature is devoted to the estimation of covariance matrices

in a large-scale setting. Recent work in this area includes the shrinkage approachproposed by Schäfer and Strimmer [16], where the authors analystically calculatethe optimal shrinkage intensity, yielding a good, computationally inexpensiveestimate. Our focus is an estimate with the property that the corresponding inversecovariance matrix is sparse.

Dempster [4] introduced the concept of covariance selection, where thenumber of parameters to be estimated is reduced by setting to zero some elementsof the inverse covariance matix. Covariance selection can lead to a more robustestimate of S if enough entries of its inverse are set to zero. Traditionally, a greedyforward/backward search algorithm is employed to determine the zero patternsLauritzen [9]. However, this method quickly becomes computationally infeasibleas p grows.2000 Mathematics Subject Classification : Primary 90C25; Secondary 90C47,49K30Keywords and Phrases : Applications of Mathematical Optimization,Robustness, Duality and Bounds, Convergence and Property of Solution, ConvexOptimization, Nesterov's Method.

1. Introduction. In this paper we investigate the following related idea.Beginning with a dense empirical covariance matrix S, we compute a maximumlikelihood estimate of S with an II-norm penalty added to encourage sparsity inthe inverse. The authors Li and Gui [10] introduce a gradient descent algorithmin which they account for the sparsity of the inverse covariance matrix by defininga loss function that is the negative of the log likelihood function. Recently, Huang,Liu and Pourahmodi [7], and Dahl [2] considered penalized maximum liklihoodestimation, and Dahl [2] in particular, proposed a set of large scale methods tosolve problems where a sparse structure of S-1 is known a priori. Our contributionis threefold: we present a provably convergent algorithm that is efficient for large-

18

scale instances, yielding a sparse, invertible estimate of S-1, even for n<p; weobtain some basic complexity estimates for the problem; and finally we test ouralgorithm on synthetic data as well as gene expression data from two datasets.Notations. For a p×p matrix X,X0 means X is symmetric and positive semi-

definite; x denotes the largest singular value norm, x 1 the sum of the absolute

values of its elements, and x their largest magnitude.

2. Preliminaries. In this section we set up the problem and discuss someof its properties.2.1. Problem Setup. Let S0 be a given empirical matrix, for data drawn from amultivariate Gaussian distribution. Let the variable X be our estimate of theinverse covariance matrix. We consider the penalized maximum-liklihood problem

maxlog det X X ,X x 1 ...(1.1)

where S,X trace( SX ) denotes the scalar product

between two symmetric matrices S and X, and the

term iji , jx X 1 penalizes nonzero elements of X.

Here, the scalar parameter >0 controls the size of the penalty, hence the sparsityof the solution. The penalty term involving the sum of absolute values of the entriesof X is a proxy for the number of its nonzero elements, and is often used-albeitwith vector, not matrix, variables-in regression techniques, such as LASSO inTibshirani [17], when sparsity of the solution is a concern.

The classical maximum likelihood estimate of is recovered for >0, andis simply S, the empirical covariance matrix. Furthermore, as noted above, forp>>n, the matrix S is likely to be singular. It is desirable for our estimate of S tobe invertible. We shall show that our proposed estimator performs someregularization, so that our estimate is invertible for every >0.2.2. Robustness, Duality and Bounds. By introducing a dual variable U, we

can write (1) as

0X U

max min log det X X ,S U .

Here U denotes the maximal absolute value of the entries of U. This corresponds

to seeking an estimate with maximal worst-case likelihood, over all componentwise bounded additive perturbations S+U of the empirical covariance matrix S.Such a "robust optimization" interpretation can be given to a number of estimationproblems, most notably support vector machines for classification.We can obtain the dual problem by exchanging the max and the min:

U

min log det S U p : U ,S U

0 (2)

The diagonal elements of an optimal U are simply iiU . The

19

corresponding covariance matrix estimate is ˆ: S U . Since the above dual

problem has a compact feasible set, the primal and dual problems are equivalent.The optimality conditions relate the primal and dual solutions by 1X .

The following theorem shows that adding the l1-norm penalty regularizesthe solution.Theorem-1 For every >0, the optimal solution to the penalized ML problem (1)is unique, and bounded as follows :

p I X p I, where pp , p .

S p

1

Proof. An optimal X satisfies X=(S+U)–1, where U . Thus, we can without

loss of generality impose that X p I , where p is defined in the theorem.

Likewise, we can show that X is bounded above. Indeed, at optimum, the primal-dual gap is zero:

1

0 log det S P p log det X S,X X

p S,X X , 1

where we have used (S+U)X=1. Since S,X are both positive semi-definite, we obtain

FX X X p I

1 as claimed. Problem (2) is smooth and convex. When

p p /1 2 is in the low hundreds, the problem can be solved by existing software

that uses an interior point method Vandenberghe [18], The complexity to computean -suboptimal solution using such-second-order methods, however, is

6 1O p log / making them infeasible for even moderately large p.

The authors Dahl et al. [2] developed a set of algorithms to estimate the nonzero

entries of 1 when the sparsity pattern is known a priori and corresponds to an

undirected graphical model that is not chordal. Here our focus is on relativelylarge, dense problems, for which the sparsity pattern is not known a priori. Notethat we cannot expect to do better than O(p3), which is the cost of solving the non-penalized problem =0 for a dense sample covariance matrix S.2.3 Choice of Regularization Parameter . In this section we provide a simpleheuristic for choosing the penality parameter , based on hypothesis testing, Weemphasize that while the choice of is an important issue that deserves a thoroughinvestigation, It is not the focus of this paper.

The heuristic is based on the observation that if ijS then there cannot be zero

20

in that element of our estimate of the covariance matrix ij 0 suppose we choose

according to

2

222

n i, j ii ji

n

t max S S ,

n t

(3)

where nt 2 denotes the two-tailed 100% point of the t-distribution, for n-2

degrees of freedom. With this choice, and using the fact that S0, it can be shown

that ijS implies the condition for rejecting the null hypothesis that variables

i and j are independent in the underlying distribution, under a likelihood ratiotest of size Muirhead [13]. We note that this choice yields an asymtoticallyconsistent estimator. As n , we recover the sample covariance S as our estimateof the covariance matrix, and S converges to the true covariance .

3. Block Coordinate Descent Methd. In this section we present anefficient algorithm for solving the dual problem (2) based on block coordinatedescent.3.1 Algorithm. We first describe a method for solving (2) by optimizing over onecolumn and row of S+U at a time. Let W:=S+U be our estimate of the truecovariance. The algorithm begins by initializing W0=S+l. The diagonal elementsof W0 are set to their optimal values, and are left unchanged in what follows.

We can permute rows and columns of W, so that we are optimizing over thelast column and row. Partition W and S as

T

W wW

w w

11 12

12 22T

S sS

s s

11 12

12 22

where pw ,s R 112 12 the update rule is found by solving the dual problem (2), with

U fixed except for its last column and row. This leads to a box-constrained quadraticprogram (QP):

Tw arg min y W y : y s

112 11 12 (4)

We cycle through the columns in order, solving a QP at each step. After each sweepthrough all columns, we check to see if the primal-dual gap is less than , a giventolerance. The primal variable is related to W by X=W–1. The duality gap conditionis then

S,X X p 1 .

3.2. Convergence and Property of Solution. Iterates produced by thecoordinate descent algorithm are strictly positive definite. Indeed, since S0, wehave that W0>0 for any >0. Now suppose that, at iteration k,W>0. This implies

21

that the following Schur complement is positive : Tw – w W w 122 12 11 12 0 . By the update

rule (4), we have 1 1

22 12 11 12 22 12 11 12 0Tˆ ˆw w W w w w W w ,

which, using Schur complements again, implies that the new iterate satisfies 0W .

Note that since the method generates a sequence of feasible primal and dual points,the stopping criterion is nonheuristic. As a consequence, the QP (4) to be solved ateach iteration has a unique solution. This implies that the method converges tothe true solution of (2), by virtue of general results on block-coordinate descentalgorithms Bertsekas [1].

The above results shed some interesting light on the solution to problem

(2). Suppose that the column s12 of the sample covariance satisfies s 12 , where

the inequalities hold component wise. Then the corresponding column of the

solution is zero 12 0 . Indeed, if the zero vector is in the constraint set of the QP

(4), then it must be the solution to that QP. As the constraint set will not changeno matter how many times we return to that column, the corresponding column ofall iterates will be zero. Since the iterates converge to the solution, the solutionmust have zero for that column. This property can be used to reduce the size of theproblem in advance, by setting to zero columns of W that correspond to columnsin the sample covariance S that meet the above condition.

Using the work of Luo and Tseng [11], it is possible to show that the localconvergence rate of this method is at least linear. In practice we have found that asmall number of sweeps through all columns, independent of problem size p, issufficient to achieve convergence. For a fixed number of K sweeps, the cost of themethod is O(Kp4), since each iteration costs O(p3).3.3 Connection to LASSO. The dual of (4) is

11 12 1T T

xmin x W x s x . (5)

Strong duality obtains so that problems (5) and (4) are equivalent. If we let Q

denote the square root of W ,b : Q 111

12

, then we can write(5) as x

min Qx b x 2

2 1

The above is a penalized least-squares problem, often referred to as LASSO.If W11 were a principal minor of the sample covariance S, then the above would beequivalent to a penalized regression of one variable against all others. Thus, theapproach is reminiscent of the approach explored by Meinshausen and Buhlmann[12]. but there are two major differences. First, we begin with some regularization,and as a consequence, each penalized regression problem has a unique solution.Second, and more importantly, we update the problem data after each regression;

22

in particular, W11 is never a minor of S. In a sense, the coordinate descent methodcan be interpreted as a recursive LASSO method.

4. Nesterov's Method. In this section, we apply the recent results due toNesterov [15] to obtain a first-order method for solving (1). Our main goal is notto obtain another algorithm, as we have found that the coordinate descent is alreadyquite efficient; rather, we seek to use Nesterov's formalism to derive a rigorouscomplexity estimate for the problem, improved over that delivered by interior pointmethods.

As we shall see, Nesterov's framework allows us to obtain an algorithm

that has a complexity of .O p / 4 5 , where >0 is the desired accuracy on the

objective of problem (1). This is to be contrasted with the complexity of interior-

point methods, O p log / 6 1 . Thus, Nesterov's method provides a much better

dependence on problem size, at the expense of a degraded dependence on accuracy.In our opinion, obtaining an estimate that is accurate numerically up to dozens ofdigits has little practical value, as it is much more important to be able to solvelarger problems with less accuracy. Note also that the memory requirements forNesterov's methods are much better than those of interior-point methods.4.1 Idea of Nesterov's Method. Nesterov's method [15] applies to a class ofnon-smooth, convex optimization problems, of the form

x

min f x : x Q 1 (6)

where the objective function is described as

u

ˆf x f x max Ax,u : u Q . 22

Here Q1 and Q2 are bounded, closed, convex sets, f x is differentiable

(with Lipschitz continuous gradient) and convex on Q1, and A is a linear operator.Observe that we can write (1) in this form if we impose bounds on the eigenvaluesof the solution, X. To this end, we let

Q X : I X I 1 ,

Q U : U

2 ,

where , 0 are given. We also define

f X log det X S,X , and A I .

To Q1 and Q2, we associate norms and continuous, strongly convex functions,

called prox-functions, d X1 and d U2 . For Q1 we choose the Frobenius norm,

and a prox-function d X log det X log 1 . For Q2, we choose the Frobenius

norm again, and a prox-function 22 2

Fd U U / .

23

The method applies a smoothing technique to the non-smooth problem (6),which replaces the objective of the original problem, f(X), by a penalized function

involving the prox-function d U2 :

U Q

ˆf X f X mix AX ,U d U

2

2 . (7)

The above function turns out to be a smooth uniform approximation. It isdifferentiable, convex on Q1, and has a Lipschitz-continuous gradient, with aconstant L that can be computed as detailed below. A specific gradient scheme isthen applied to this smooth approximation, with convergence rate O(L/).4.2 Algorithm and Complexity Estimate. To detail the algorithm and computethe complexity, we first calculate some parameters corresponding to our definitions

above. First, the strong convexity parameter for d1(X), on Q1 is 21 1 / , in the

sense that Fd X H,H trace X HX H H 22 1 1 2

1 for every symmetric H.

Furthermore, the center of the set Q1 is X QX arg min d X I 10 1 , and satisfies

d X 1 0 0 . Without choice, we have x QD max d X log / 11 1 .

Similary, the strong convexity parameter for d2(U) on Q2 is 2 1 and we

have 2

22 2 2U QD max d U p / . With this choice, the center of the set Q2 is

U QU arg min d U . 20 2 0

For a desired accuracy , we set the smoothness parameter / D 22 , and start

with the initial point X I 0 . The algorithm proceeds as follows.

For k 0 do

1. Compute *k k kf X X S U X 1 where *U X solves (7).

2. Find k k k k FY arg min f X ,Y X L Y X : X Q

2

1

12

.

3. Find 1 101

12

k

k X i ii

L iZ argmin d X f X ,X X : X Q

.

4. Update k k k

kX Z Y

k k

2 1

3 3.

In our case, the Lipschitz constant for the gradient of our smooth

approximation to the objective function is L M D A / 22 22 , where

24

M / 21 is the Lipschitz constant for the gradient of f , and the norm A is

induced by the Frobenius norm, and is equal to . The algorithm is guaranteed toproduce an -suboptimal solution after a number of steps not exceeding

D D MDN A

1 2 1

1 2 1

14 (8)

k p log kp

4 ,

where k / is bound on the condition number of the solution.

Now we are ready to estimate the complexity of the algorithm. For step 1,the gradient of the smooth approximation is readily computed in closed form, viathe computation of the inverse of X. Step 2 essentially amounts to projecting onQ1, and requires an eigen value problem to be solved; likewise for step 3. In fact,

each iteration costs O p3 . The number of iterations necessary to achieve an

objective with absolute accuracy less than is given in (8) by, N O p / 15 , ifthe condition number k is fixed a priori. Thus, the Complexity of the algorithm is

.O p / 4 5 .

5. Numerical Results. In this section we present some numerical results.We begin with a small synthetic example to test the ability of the method to recovera sparse structure from a noisy matrix. Starting with a sparse matrix A, we obtain

S by adding a uniform noise of magnitude . 0 1 to A1 .

In figure 1 we plot the sparsity patterns of A, S–1, and the solution X to (1)

using S and .

Figure 1. Recovering the Sparsity PatternWe plot the underlying sparse matrix A, the inverse of the noisy version of A-1,and the solution to problem (1) for equal to the noise level.We next perform the following experiment to see what happens to the solution of(1) as we vary the parameter above and below the noise level . For each value of

A S–1 X

25

, we randomly generate 10 sparse matrices A of size n=50. We then obtain samplecovariance matrices S as above, again using =0.1.

Figure 2. Recovering StructureGene Expression PropertiesThe 300 experiment compendium dataset contains n=253 samples with p=6136variables. With a view towards obtaining a very sparse graph, we set =0.1 in theheuristic formula (3) of section (2.3) to obtain =0.0313.Applying the property of the solution discussed in section (3.2), the size of the

problem was reduced to p =537. Three sweeps through all columns were required

to achieve a duality gap of . 0 146 , with a total computing time of 18-minutes,

34-seconds. The resulting estimate of the inverse covariance matrix ˆ 1 is 99%

sparse and has a condition number of 21.84. Figure (4) shows a sample subgraph

obtained from ˆ 1 , generated using the Graph Explore program developed by Dobra

and West [6]. The method has picked out a cluster of genes associated with aminoacidmetabolism, as described by Hughes et al. [7].

Figure 3. Application to Hughes Dataset Using =0.0313.As we have seen, the penalized maximum likelihood problem formulated here is

useful for recovering a sparse underlying precision matrix 1 from a dense sample

covariance matrix S, even when the number of samples n is small relative to thenumber of variables p. In preliminary tests, the method appears to be a potentiallyvaluable tool for analyzing gene expression data, although further testing is

7

6

5

4

3

2

1

-0.5 0 0.5 1

��

��

��

�

��

26

required.REFERENCES

[1] D. Bertsekas, Non linear programming, Athena Scientific (1998).[2] J. Dahl., V. Roycho Wdhury and L. Vandenberghe, Maximum Likelihood Estimation of

Gaussian Graphical Models : Numerical Implementation and Topology Selection, UCLAPreprint.

[3] A. D' Aspremount, L. El Ghaoui, M. Jordan and G.R.G. Lanckriet, A direct formulationfor sparse PCA using semide. nite programming. Advances in Neural InformationProcessing Systems, 17 (2004).

[4] A.P. Dempster, Covariance selection, Biometrics, 28 (1972), 157-175.[5] A. Dobra, C. Hans, B. Jones, J.J.R. Nevins, G. Yao and M. West, Sparse graphical

models for exploring gene expression data, Journal of Multivariate Analysis, 90 (2004),196-212.

[6] A. Dobra and M. West, Bayesian covariance selection, Working paper, ISDS, DukeUniversity, 2004.

[7] I.Z., Huang, N., Liu, and M. Pourahmadi, Covariance selection and estimation viapenalized normal likelihood, Wharton Preprint, 2005.

[8] S.H. Functional discovery via a compendium of expression Pro. Les. Cell, 102 (2000),109-126.

[9] S. Laurizen, Graphical Models. Springer Verlag, 1996.[10] H. Li and I. Gui, Gradient directed regularization for sparse Gaussian concentration

graphs, with applications to inference of genetic networks, University of PennsylvaniaTechnical Report, 2005.

[11] Z.Q., Luo, and P. Tseng, On the convergence of the coordinate descent method for convexdifferentiable minimization, Journal of Optimization Theory and Applications, 72, (1992)7-35.

[12] N. Meinshausen and P. Buhlmann. High dimensional graphs and variable selectionwith the lasso, Annals Statistics, 2005 (in press)

[13] R.J. Muirhead, Aspcts of Multivariate Statistical Theory, John Wiley and Sons, Inc,1982.

[14] G. Natsoulis, L., El Ghaoui, G. Lanckriet, A. Tolley, F. Leroy, S. Dunlea, B. Eynon, C.Pearson, S. Tugendreich and K. Jarnagin, Classification of a large microarray data set:algorithm comparison and analysis of drug signatures, Genome Research, 15 (2005),724-736.

[15] Y., Nesterov, Smooth minimization of non-smooth functions, Math. Prog. Ser. A, 103(2005), 127-152.

[16] J., Schafer, and K. Strimmer, A shrinkage approach to large-scale covariance matrixestimation and implications for functional genomics, Statistical Applications in Geneticsand Molecular Biology, 4 (2005).

[17] R. Tibshirani, Regression shrinkage and selection via the lasso, Journal Royal StatisticalSociety, Series B, 58 (1996).

[18] L. Vandenberghe, S. Boyd, and S.-P. Wu, Determinant maximization with linear matrixinequality constraints. SIAM Journal on Matrix Analysis and Applications, 19 (1998),499-533.

27


ON DECOMPOSITION OF CURVATURE TENSOR FIELDS IN AKAEHLERIAN RECURRENT SPACE OF FIRST ORDER

ByK.S. Rawat and Nitin Uniyal

Department of Mathematics, H.N.B. Garhwal University CampusBadshahi Thaul, Tehri (Garhwal)-249199, Uttarakhand, India

E-Mail : [email protected] and [email protected]

(Received : June 15, 2010; Revised : September 20, 2011)

ABSTRACTTakano [13] studied decomposition of curvature tensor in a recurrent space.

Sinha and Singh [12] defined and studied defined decomposition of recurrentcurvature tensor field in a Finsler space. Negi and Rawat ([2][3],[4]) studieddecomposition of recurrent curvature tensor field in a Kaehlerian space. Rawat[5], Rawat and Silswal ([6],[7]) defined and studied decomposition of recurrentcurvature tensor fields in a Tachibana space. Further, Rawat and Dobhal ([8],[9])studied decomposition of recurrent corvature tensor field in a Kaehlerian recurrentspace. Rawat and Singh ([10],[11]) studied the decomposition of curvature tensorfields in a Kaehlerian recurent space of first order.

In the present paper, we consider the decomposition of curvature tensorfield Rhi

jk in terms of two non-zero vectors and a tensor field. Also several theoremsare established and proved therein.2000 Mathematics Subject Classification : 53A45Keywords and Phrases : Decomposition, Kaehlerian, Recurrent, Projectivecurvature tensor.

1. Introduction. When in a 2n-dimensional real space X2n of classrC ( r ),2 there is a mixed tensor field h

i ,i , jF ;R , , ,..., n 1 2 3 2 satisfying

h h hi l jF F A , ...(1.1)

we say that the space admits an almost complex structure and we call such a spacean almost complex space.

If an almost complex space has a positive definite Riemannian metricj i

jids g d d 2 which setisfiesl kj i lk jiF F g g , ...(1.2)

Then the space is called an almost-Hermitian space.

In this case the tensor ih

defl

i lhF F g is anti-symmetric(or skew-symmetric) in i and h.

28

If an almost-Hermitian space satisfies 0j ih i hj h ji F F F , ...(1.3)

where j denotes the operator of covariant differentiation with respect to the

metric tensor gji of the Riemannian space then it is called an almost-Kaehlerianspace and if it satisties

0j ih i jhF F , ...(1.4)

then it is called a K-space.In an Almost-Hermitian space, if

0 0j ih ih,j F , or F , ...(1.5)

then it is called a Kaehlerian space.The Riemannian curvature tensor field is defined by

h h h h l h lijk i jk j ik il jk jl ikR , ...(1.6)

where ii / x and ix denotes the real local coordinates.

The Ricci tensor and the scalar curvature are given bya

ij aijR R and ijijR R g respectively.

It is well known that these tensors satisfy the following identitiesaijk,a jk,i ik, jR R R ...(1.7)

a,i i ,aR R 2 ...(1.8)

a ai aj ia jF R R F ...(1.9)

and a j a ji a i aF R R F . ...(1.10)

The holomorphically projective curvature tensor is defined by

h h h h h h hijk ijk ik j jk i ik j jk i ij kP R R R S F S F S F ,

n

1

22 ...(1.11)

where aij i ajS F R .

The Bianchi identities in Kn are given by

0h h hijk jki kijR R R , ...(1.12)

and 0h h hijk,a ika , j iaj ,kR R R . ...(1.13)

The commutative formulae for the curvature tensor field are given as follows

i i a i, jk ,kj ajkT T T R ...(1.14)

29

and h h a h h ai,ml i,lm i aml a imlT T T R T R . ...(1.15)

A Kaehlerian space Kn is said to be Kaehlerian recurrent space of first order, if itscurvature

h ha ijk a ijk R R ,

i.e. h hijk,a a ijkR R , ...(1.16)

where a is a non-zero vector and is known as recurrent vector field. The space issaid to be Ricci-recurrent space of first order, if it satisfies the condition

ij ,a a ijR R , ...(1.17)

Multiplying the above equation by gij, we have

,a aR R. ...(1.18)

2. Decomposition of Curvature Tensor Field hijkR . We consider the

decomposition of recurrent curvature tensor field hijkR in the following form

h hijk i j ,kR v' ...(2.1)

where two vectors u'h, i and tensor field j ,k are such that

hhv' . 1 ...(2.2)

Theorem 2.1. Under the decomposition (2.1), the Bianchi identities for hijkR take

the forms

i j ,k j k,i k i, j 0 ...(2.3)

and a j ,k j k,a k a , j 0 . ...(2.4)

Proof. From equations (1.12) and (2.1), we have

i j ,k j k,i k i, j 0 . ...(2.5)

Since hv' 0 .

From equations (1.13), (1.16) and (2.1), we get

hi a j,k j k,a k a, jv' 0 . ...(2.6)

Multiplying (2.6) by h and using (2.2), we obtain

0i a j ,k j k,a k a, j . ...(2.7)

Since i , 0 therefore, we derive

30

a j ,k j k,a k a , j 0 .

This completes the proof of the theorem.

Theorem 2.2. Under the decomposition (2.1), The tensor field hijk ijR ,R and j ,k

satisfy the relationsa

a ijk i jk j ik i j ,kR R R ...(2.8)

Proof. With the help of equations (1.7), (1.16) and (1.17), we havea

a ijk i jk j ikR R R ...(2.9)

Multiplying (2.1) by h and using relation (2.2), we get

hh ijk i j ,kR ...(2.10)

From equations (2.9) and (2.10, we derive the required relation (2.8).

Theorem 2.3. Under the decomposition (2.1), the quantities a and hv' behave

the recurrent vectors. The recurrent form of these quantities are given by

a,m m a ...(2.11)

and ,h h,m mv v' ...(2.12)

Proof. Differentiating (2.8) covariantly w.r.t. mx and using (2.1) and (2.8), we obtain

ha ,m i j ,k i ,m jk j ,m ikv' R R . ...(2.13)

Multiplying (2.13) by a and using (2.1) and (2.9), we have

a,m i jk j ik a i,m jk j,m ikR R R R , ...(2.14)

Now, multiplying equation (2.14) by h we get

a,m i jk j ik h a h i,m jk j,m ikR R R R . ...(2.15)Since the expression on the right hand side of above equation is symmetric in aand h, therefore

a,m h h,m a , ...(2.16)

provided that

i jk j ikR R . 0

The vector field a being non-zero, we can have a proportional vector m such

that

a,m m a ...(2.17)

Further, differentiating the equation (2.2) w.r.t. mx , we have

,h hh ,m h,mv v' . 0

31

Making use of equation (2.11), we derive,h h,m mv v' (since h 0 ). ...(2.18)

This proves the theorem.

Theorem 2.4 Under the decomposition (2.1), the vector i and the tensor j ,k

satisfy the equation i j ,k m m i j ,km j,k i,m , ...(2.19)

where j,km j,k,m.

Proof. Differentiating (2.1) covariantly w.r.t. xm and using equations (1.16), (2.1)and (2.12), we get the required result of the Theorem.Theorem 2.5 Under the decomposition (2.1), the curvature tensor andholomorphically projective curvature tensor are equal if

2 0h h h l h l h lk,m i j j i l j i i j l j ,m k iF F F F F F . ...(2.20)

Proof. The equation (1.11) may be written in the formh h h

ijk ijk ijkP R D , ...(2.21)

where

1

22

h h h h h hijk ik j jk i ik j jk i ij kD R R S F S F S F .

n ...(2.22)

Contracting indices h and k in (2.1), we havek

ij i j ,kR v' . ...(2.23)

In view of equation (2.23), we get

l mij i l j,mS F v' . ...(2.24)

Making use of relations (2.23) and (2.24) in equation (2.22), we have

h m h h h l h l m h lijk k,m i j j i l j i i j l j ,m k iD v' F F F F v' F F

n

12 0

2 ...(2.25)

From (2.21), it is clear that h hijk ijkP R if h

ijkD 0 ,

which, in view of (2.25), becomes

m h h h l h l m h lk,m l j j l l j i i j l j ,m k iv ' F F F F v' F F 2 0 ...(2.26)

Multiplying the above equation by m and using relation (2.2), we obtain the

required result (2.20).Theorem 2.6 Under the decomposition (2.1), the scalar curvature R, satisfies therelation

32

ijk ,k i j ,kR R g .

Proof. Contracting indices h and k in (2.1), we havek

ij i j ,kR v' . ...(2.27)

Multiplying (2.27) by gij both sides, we haveij k

i j ,kR g v' . ...(2.28)

Multiplying (2.28) by k and using (2.2), we getij

k i j ,kR g

or, ij,k i j ,kR g [by using (1.18)] .

REFERENCES[1] K.B. Lal and S.S. Singh, On Kaehlerian spaces with recurrent Bochner curvature, Atti

Accad. Naz. Lincei Rend. Cl. Fis. Mat. Natur. (Ser.8) 51 (1971), 213-220.[2] D.S. Negi and K.S. Rawat, Decomposition of recurrent curvature tensor fields in a

Kaehlerian recurrent space, Acta Cien, lnd. 21M, No.2, (1995), 139-142.[3] D.S. Negi and K.S. Rawat, On decomposition of recurrent curvature tensor fields in a

Kaehlerian space, Acta Cien.lnd. 21M, No.2, (1995), 151-154.[4] D.S. Negi and K.S. Rawat, The study of decomposition in a Kaehlerian space, Acta Cien.

Ind. 23M, No.4, (1997), 304-311.[5] K.S. Rawat, The decomposition of recurrent curvature tensor fields in a Kaehlerian

space, Acta cien. Ind., 21M, N0.1, (2005), 135-138.[6] K.S. Rawat and G.P. Silswal, Decomposition of recurrent curvature tensor fields in a

Kaehlerian recurrent space, Acta Cien. Ind., 31 M, No.3, (2005), 795-799.[7] K.S. Rawat and G.P. Silswal, Decomposition of recurrent curvature fields in a Tachibana

space, Jour. PAS, 13 (Ser. A), (2007), 13-17.[8] K.S. Rawat and Girish Dobhal, Some theorems on decomposition in a Tachibana recurrent

space, J. PAS, 13 (Ser. A) (2007), 458-464.[9] K.S. Rawat and Girish Dobhal,Study of the decomposition of recurrent curvature tensor

fields in a Kaehlerian recurrent space, Jour. Pure and Applied Mathematika Sciences,68 No. 1-2 Sept (2008), 83-88.

[10] K.S. Rawat and Kunwar Singh, The study of decomposition of curvature tensor fields ina Kaehlerian recurrent space of first order, J. Tensor Society, 3 (2009), 11-18, MR 2567069.

[11] K.S. Rawat and Kunwar Singh, The study of decomposition in a Tachibana recurrentspace, Jour. PAS, 15 (Mathematical Sciences) (2009), 430-436.

[12] B.B. Sinha and S.P. Singh, On decomposition of recurrent curvature tensor fields in aFinsler space, Bull. Cal. Math. Soc., 62 (1970), 91-96.

[13] K. Takano, Decomposition of curvature tensor in a recurrent space, Tensor (N.S.), 18,No.3 (1967), 343-347.

[14] K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press(1965).

33


FOURIER SERIES FOR MODIFIED MULTI-VARIABLE H-FUNCTIONBy

Rashmi Singh

Department of MathematicsB.N. Postgraduate Girls College, Udaipur-313001, Rajasthan, India

E-Mail : [email protected]

R.S. Chandel

Former Head, Department of Mathematics and StatisticsM.L. Sukhadia University, Udaipur-313001, Rajasthan, India

(Received : June 25, 2010)

ABSTRACTThe object of this paper is to derive integral involving modified H-function

of several variables. These integrals are used to establish the Fourier series forgenerlized function. By suitably specializing the coefficients and the parameterswe can obtain many (new and known) interesting results involving multi-variableH-functions [8]. The results obtained by Srivastava and Panda [8], Kaul [1,2], MacRobert [3] and Sneddon [7] follow as particular cases of our results.2000 Mathematics Subject Classification : 33C70, 42A24Keywords and Phrases : H-function, Multi-variable H-function, Modified Multi-variable H-function.

1. Introduction. The modified multi-variable H-function employed askernel of multi-dimentional transform defined by Prasad and Singh [6] on thelines of Srivastava and Panda [8], Prasad and Maurya [5] is as follows:

11 1

1 1

1

111 1 1

11 1 1

rr r

r r

r

r r r r rj j j j j j j j j j j j,p,p ,IR' ,pm,n: R':m ,n ,;...;m ,n

p,q: R: p q ;...; p ,q r r r r rj j j j j j j j j j j j,q,q ,IR ,qr

Z a ; ,..., : e : u g ,...,u g : c , ;...; cH

b : ,..., : l ;U f ,...,U f : d , ;...; dZ

1

1

1 1 1 1 1

1

2r

r

r r r r rrL L

... ... ,..., z ...z d ...d ...(1.1)

where

1 1

1 1

1

1

i i

i i

i i

m ni i ij i j j i

j ji i q p

i i i ij j i j j i

j m j n

d c

d c 1 2i , ,...,r ...(1.2)

34

1 11 1

1

1 11 1

1

1

r rm ni i

j j i j j ij ji i

r q r p ri i ij j i j j i

j m j ni i

b a,...,

b a

1 1

1 1

rIR'i i

j j j ij i

rIR'i i ij j j i

j i

e u g.

I U f

...(1.3)The multiple integral (1.1) converges absolutely if

12i iarg z U , i , ,...,r . 1 2

where

i i i

i

m q nq pm ni i i i i i i

i j j j j j j jj j m j j n j j m j

U

1 1 1 1 1 1 1

i

i

p IR' IRi i ij j j

j n j j

g f

1 1 1

0 i , ,...,r . 1 2

We shall require the following known integrals.

1. 12 2

02 2cosP cos / sin / d

113 2

1

1 2 1 21 2 1 21

1 2 1 22 1/ , P, P / ;P / /

FP / , / ;P

...(1.4)

provided that Re 2 1 0 , Re 1 1 0 and P , , ,... 0 1 2 .

2. 12 2

02 1sin s cos sin d

1

1

2 2 1 2 13 2 2 1

s s /s / s

13 2

1 1 21

1 2 3 2, s, s / ;

F ,s / , / ;

provided that Re 2 1 0 , Re 1 1 0 and s , , ,... 0 1 2 .

2. Main Integrals.

(I)

1 1 12 2 2 2 2 2

10 2 2 2 2 2 2

r rh k h k

rcos sin H cos sin z ,..., cos sin z d

35

1 1

1 1

11 1 12

2 11 1 1

1 2 1 2r r

r r

r rm,n :IR,m ,n ;...;m np ,q :IR ',p ,q ;...; p q

r rr

z/ ; h ,...,h , / ; k ,...,k ,

H; h k ,...,h k ,

z

1

1

11 1 1

11 1 1

r

r

r r r r rj j j j j j j j j j j j,p,p ,IR' ,p

r r r r rj j j j j j j j j j j j,q,q ,IR ,q

a ; ,..., : e : u g ,...,u g : c , ;...; c ,

b : ,..., : l ;U f ,...,U f : d , ;...; d ...(2.1)

provided that1

2 2 0r

i ii

Re h

, 1

1

2 2 1 0r

i ii

Re k

,

where i ii j jmin Re d ij ,...m ,i ,...,r 1 1

(iii) 0along with the conditions for the convergence of modified H-function.

(II)

1 1 12 2 2 2 2 2

10 2 2 2 2 2 2

r rh k h k

rcosP cos sin H cos sin z ,..., cos sin z d

1 1

1 1

22 1

0

1 1 21 2

r r

r r

NPm ,n :IR ':m ,n ,...,m ,nN Np ,q :IR: p ,q ,..., p ,q

N N

P P /H

/ N !

111 1 1

1 1 11

1 2 1 2

rj j j ,pr r

rr r j j j ,qr

z a ; ,..., :/ N P ;h ,...,h , / N ;k ,...,k ,P ;h k ,...,h k , b : ,..., :z

1

1

11 1

11 1

r

r

r r r rj j j j j j j j j,p,IR' ,p

r r r rj j j j j j j j j,q,IR ,q

e : u g ,...,u g : c , ;...; c ,

l ;U f ,...,U f : d , ;...; d , ...(2.2)

with the conditions of (2.1) and P=0,1,2,... .

(III) 1 1 12 2 2 2 2 210

2 1 r rh k h krsin P cos sin H cos sin z ,..., cos sin z d

1 1

1 1

22 1

0

1 1 22 1

3 2r r

r r

NPm,n :IR':m ,n ,...,m ,nNp ,q :IR: p ,q ,...,p ,q

N N

P P / NP H

/ N !

36

111 1 1

1 1 11

1 21 2

rj j j ,pr r

rr r j j j ,qr

z a ; ,..., :/ N P ;h ,...,h , N ;k ,...,k/ P ;h k ,...,h k b : ,..., :z

1

1

11 1

11 1

r

r

r r r rj j j j j j j j j,p,IR' ,p

r r r rj j j j j j j j j,q,IR ,q

e : u g ,...,u g : c , ;...; c

l ;U f ,...,U f : d , ;...; d ...(2.3)

provided thatr

i ii

Re k

1

1

1 0 ,

where 1 2 1 2i ii j j imin Re d / , j , ,...,m ,i , ,...,r and P , , ... 0 1 2

3. Proof. In order to prove (2.1), we substitute for the modified multi-variable H-function in terms of its contour integral of Mellin-Barenes type andchange the order of integration (which is permissible). We then evaluate the innerintegral using Gamma function. On interpreting the resulting contour integralby means of (1.1), we obtain (2.1). Proceeding in the similar way, using (1.4) and(1.5) we obtain (2.2) and (2.3).

4. Fourier Series. We shall now establish following Fourier series formdified multi-variable H-functions :

1 1 12 2 2 2 2 2

12 2 2 2 2 2r rh k h krcos / sin / H cos / sin / z ,..., cos / sin / z

1 1

1 1

1111

1 1

11 1 1

11

2r r

r r

rj j j ,prm,n :IR':m ,n ;...;m ,n

p ,q :IR:p ,q ;...; p q rj j j ,qr rr

z a ; ,..., :;h ,...,h ,H

b ; ,..., :;h k ,...,h k ,z

1

1

11 1

1 011 1

1 221 2

r

r

r r r rpj j j j j j j j j,p,IR ,p s s

r r r rP s sj j j j j j j j j,q,IR ,q

e : u g ,...,u g : c , ;...; c , P P /

/ s!:U f ,...,U f : d , ;...; d ,

1 1

1 1

11 12 1

2 11 1 1

1 21 2r r

r r

rm,n :IR':m ,n ;...;m ,n rp ,q :IR: p ,q ;...; p ,q

r rr

z/ s ; k ,...,k :/ s P ; h ,...,h ,

HP ; h k ,...,h k :

z

37

r

r

r r r rrj j j j j j j j jj j j ,p,IR ,p,p


e : u g ,...,u g : c , ;...; c ,a ; ,..., :

b ; ,..., : : U f ,...,U f : d , ;...; d ,

1

1

11 11

11 1 1 ...(4.1)

with the conditions as in (2.1) and P=0,1,2,...5. Proof. In order to prove (4.1), let us suppose

12 22 2f cos / sin / ...(5.1)

1 12 2 2 2

12 2 2 2r rh k h krH cos / sin / z ,..., cos / sin / z

pP

C C cosP

01

12 . ...(5.2)

Integrating (5.1) between the limits 0 to and using the reult (2.3), we

obtain the value of C0 . Again multiplying both the sides of (5.1) by cos P and

integrating from 0 to w.r.t. , using (2.2) we get

1 1

1 1

22 1

0

1 1 221 2

r r

r r

m,n :IR':m ,n ;...;m ,ns s sp p ,q :IR:p ,q ;...; p ,q

s s

P P /C H

/ S!

11 1

1 1 1

1 2 1 2r r

r rr

z/ s P ;h ,...,h , / s ;k ,...,k :P ;h k ,...,h k :

z

r

r

r r r rrj j j j j j j j jj j j ,p,IR ,p,p


e : u g ,...,u g : c , ;...; c ,a ; ,..., :

b ; ,..., : : U f ,...,U f : d , ;...; d ,

1

1

11 11

11 1 1 ...(5.3)

with the conditions as in (2.1). Putting the values of C0 and Cp in (5.2). We obtain(4.1). Fourier sine series may also be obtained similarly.

6. Special Cases.Case I. If we put m IR' IR 0 in (2.1) we get result in terms of multi-variable

H-function defined by Srivastava and Panda [8] as :

1 1 12 2 2 2 2 210

2 2 2 2 2 2r rh k h krcos / sin / H cos / sin / z ,..., cos / sin / z d

1 1

1 1

111 1 10 2

2 11 1 1

1

1 2 1 2r r

r r

rj j j ,pr r,n :m ,n ;...;m ,n

p ,q : p ,q ;...; p ,q rr r j j j ,qr

z a ; ,..., :/ ;h ,...,h , / ;k ,...,kH

;h k ,...,h k , b ; ,..., :z

38

1

1

1 1

1 1

r

r

r rj j j j,p ,p

r rj j j j,q ,q

c ; ;...; c ,

d , ;...; d ,

provided that

1.1

2 2 1 0r

i ii

Re h

,

2. 11

2 2 1 0r

i ii

Re k

where

1 i

i ii j jj m

min Re d

(i=1,2,...,r) .

Case II. Similarly if we put m=IR'=IR=0 in (4.1) we get the result in terms ofmultivariable H-function defined by Srivastava and Panda [8] as:

1 1 12 2 2 2 2 212 2 2 2 2 2r rh k h k

rcos / sin / H cos / sin / z ,..., cos / sin / z

1 1

1 1

1110 1

1 11 1 1

1

1 21 r r

r r

rj j j ,pr,n :m ,n ;...;m ,n

p ,q : p ,q ;...; p ,q rr r j j j ,qr

z a ; ,..., :/ ;h ,...,h ,H

;h k ,...,h k , b ; ,..., :z

1

1

1 1

1 01 1

1 221 2

r

r

r rpj j j j,p ,p s s

r rP s sj j j j,q ,q

c , ;...; c , P P /

/ s!d , ;...; d ,

1 1

1 1

11 10 2 1

2 11 1 1

1 21 2r r

r r

r,n :m ,n ;...;m ,n rp ,q : p q ;...; p q

r rr

z/ s ;k ,...,k :/ s P ;h ,...,h ,

HP ;h k ,...,h k :

z

r

r

r rrj j j jj j j ,p ,p,p

r r rj j j j j j j,q,q ,q

c , ;...; c ,a ; ,..., :

b ; ,..., : d , ;...; d ,

1

1

1 11

11 1

, ...(6.2)

with the conditions as in (6.1) and P=0,1,2,...

Case III. Again in (1.4) if we put 1 0 , then the hyper-geometric function F3 2

reduces to F2 1 . The resultant is written as:

39

2 1 1 0

c c a bF a,b,c; ,Re c a b

c a c b .

On further simplification, we get

2

0

2 12

2 1cos P cos / d

P , ...(6.3)

provided that 1 2Re / and P=0,1,2,... which is known result given by

MacRobert [3].

Case IV. Also if we put 0 in (1.4) and proceed similarly as above, we obtain

1

1

2 120

1

2 1 1 22

2 1/ P

cosP sin / dP , ...(6.4)

provided that 1 1 2Re / and P , , ,... 0 1 2 which is the known result due to

Sneddon [7].Case V. In (2.2) using r=2 and putting p1=k1=k2=.....=kr=0. we obtain a result interms of H-function of two variables (defined by Mittal and Gupta [4]), given bykoul [1,2].Case VI. Again using 1=h1=h2=.....=h1=k1=k2=...=kr=0 and taking r=2 in(2.2) we get the result given by Koul [1.2].

REFERENCES[1]. C.L. Koul, Certain properties of Fourier Kernels, Math. Education Sect. A 4(1970), 39-

49.[2]. C.L. Koul, Fourier series of a generalized function of two variables, Proc. Indian Acad,

Sci. Sect. A 75 (1972), 29-38.[3]. T.M. Mac Robert, Fourier Series for E-function, Math. Z., 75 (1961), 79-82.[4]. P.K. Mittal and K.C. Gupta, An integral involving generalized function of two variables,,

Proc.Indian Acad. Sci. Sect. A, 75 (1972), 117-123.[5]. Y.N. Prasad and R.P. Maurya, Basic properties of the generalized multiple L-H transform,

Vijnan Prishad Anusandhan Patrika, 22 No. 1 (Jan 1979), p. 74.[ 6]. Y.N. Prasad and A.K. Singh, Basic properties of the transform involving H-function of

r-variables as kernel, Indian Acad. Math. 4 No. 2 (1982), 109-115.[7]. I.N. Sneddon, Special Functions of Mathematical Physics and Chemistry, Inter. Science

Publishers, New-York, (1965), 41.[8]. H.M. Srivastava, and R. panda, Expansion theorms for the H-function of several complex

variables, J. Reine Angew, Math. 288, (1976), 129-145.

41


PHASE SHIFTS OF S-WAVE SCHRÖDINGER EQUATION FORMITTAG- LEFFLER FUNCTION

ByHemant Kumar

Department of Mathematics, D.A-V. Postgraduate College, Kanpur-208027Uttar Pradesh, India


Vimal Pratap Singh

Department of Mathematics U.I.E.T. Building, C.S.J.M. University,Kanpur-208027, Uttar Pradesh, IndiaE-Mail:[email protected]

ABSTRACTIn this paper, we consider a potential function to study the phase shift

difference of s-wave Schrödinger equation.2000 Mathematics Subject Classification : Primary 33C20, Secondary 33E1265D20.Keywords: s-wave Schrödinger equation, Phase shifts, Mittag-Leffler function.

1. Introduction. The phase shifts have great importance in thecomputation of scattering and experimental work of nuclear and atomic collision.Several mathematical problems are solved to find out a potential function from aobserved phase shift difference. Many workers namely Mahajan and Varma [8],Raghuwansi and Sharma [9], Kumar, Chandel and Agrawal [5], Agrawal and Kumar[1], Chandel and Kumar [3] have determined the phase shifts from a given potentialfunction. Recently, Kumar and Singh [6] have obtained an approximation formulafor phase-shifts of s-wave Schrödinger equation with the application of binomialpotential function and study the phase-shifts variation with respect to theparameter involving in binomial potential function.

Here, in our work, we consider a potential function of Mittag- Lefflerfunction and then on applying Tietz method [12], we obtain phase shift differencein series form involving hypergeometric function [8] and then make some studieson shift variation verses the parameters involving in the potential function.

The Mittag- Leffler function (see Erdélyi et al. [4] and Srivastava andManocha [11]) is defined by

n

an

zE z ,a C

an

0 1 . ...(1.1)

42

We consider the potential function in the form

braU r E e , a,b C . ...(1.2)

Particularly, on setting a=0 in (1.2), we find the potential function of Bhattacharjieand Sudarshan [2] in the form

nbr

n

U ' r U r e

1

1 . ...(1.3)

Also, from (1.3), we get

1

nbrn

n

dU ' r dU rA e

dr dr at a=0 . ...(1.4)

The potential function considered in (1.2) have more parameters to that ofBhattacharjie-Sudarshan [2] function and Chandel-Kumar [3] function. In thispaper, we determine the phase shift difference formula for this function on makinguse of the Tietz' s techniques and make some studies on shift variation with respectto the parameters involving in the parameters given in the potential function(1.2).

2. Formulae Used. In our investigation, we need the applications offollowing formulae :

For the s-wave radial Schrödinger equation

22

2 2

10r

d r L LK U r

dr r . ...(2.1)

Using Tietz [12] method and the Luke [7] formulae, Chandel and Kumar [3] havegiven the phase shifts in the form

2 1

2 1 2 21 1 22 3 0

25 2 2 32 3 2 5 2

LL

L L L

L ;K dUr F K r dr,

L / , L ;L / L / dr

provided that (2L+3)>0 ...(2.2)The second phase difference formula is given by

2 1

2 3 2 21 1 1 22 3 0

13 2 2 22 1 2 3 2

LL

L L L

L ;K dUr F K r dr

L / , L ;L / L / dr (2.3)

3. Phase Shift Difference for Mittag-Leffler Function . From (2.1),we find

1

nbr

n

dU b e .

dr a an

...(3.1)

43

Now, making an appeal to (3.1) and (2.2), we derive

2 1 2

1 2 12 3 2 4 2 21

2 22 1 42 33 2

L

L L L Ln

L ,L ;K L KF

L ;ab L / n an n b ,

provided that 22 1iK / nb . ...(3.2)

Further, Making an appeal to (3.1) and (3.2), we get

2 1 2

1 1 2 12 1 2 2 2 21

1 11 1 42 21 2

L

L L L Ln

L ,L ;K L KF ,

L ;ab L / n an n b

provided that 22 1iK / nb . ...(3.3)

4. Particular Cases . For L=0 , (3.2), gives

2

0 1 2 13 4 2 21

2 21 433 2 n

, ;K KF .

;ab / n an n b ...(4.1)

while for L=1, (3.2) gives

3 2

1 2 2 15 6 2 21

3 32 1 455 2 n

, ;K KF .

;ab / n an n b ...(4.2)

From (4.2), we may write

52 2 2

1 2 3 41 1

42 213 2 3 4

s s

n s s

K / n bsKab / n an s!

. ...(4.3)

Now, adding (4.1) and (4.3) we can write

2

0 2 2 143 2 21

2 21 443 2 n

, ;K KF

;ab / n bn an

. ...(4.4)

lf we put L=1 in the equation (3.3), then we get directly above result (4.4).5. Analysis. To analize above results we start with

Table No. 1

a 0 2

1 22.92260332 11.409451073 3.8003126494 0.950005558

44

Figure 1.The above graph shows that when potential function Figune vo.1 woar is

reduced on making increment of a so that internal kinetic energy of the particle isincreased and by this kinetic energy the particle of s-orbit gains frequency of d-orbit and then phase of s-wave tends to phase of d-wave ( 0 2 ), thus differencein phase in sharply reducing.

Table No. 2

b 0 2

1 22.92260332 0.2391397053 0.0854126474 0.027040942

Variation 0–2 of with respect to b.

Figure -2

45

By second graph it is shown that when we make the increment in b, thepotential energy of the particle is grown up, as when as the potential energy ischanged, the internal energy of the particle is also changed so that the particle isdisturbed due to slightly scattering and thus phase shifts is sharply decreased butwhen potential energy becomes large the difference in phase shifts becomes constantand the particle remains is s-orbit.

Table No. 3

b 0 2

1/2 3458.013491/3 61372.553741/4 465550.48091/5 2233824.794

Variation of 0 2 with respect to b.

Figure. 3From third graph, we have that when we reduce the value of b in the formula

(1.2), the potential energy of the particle is reduced and thus internal kinetic energyof the particle is increased and by this effect the particle is scattered so that thedifference in phase is sharply increasing.

REFERENCES[1] R.D. Agrawal and H. Kumar, The phase shift difference for binomial potential function,

Jour. Maulana Azad Colloge of Tech., 32 (1999), 67-75.[2] A. Bhattacharjie and E.C.G. Sudarshan, A class of solvable potentials, Nuovo. Cim,

25 (1962), 864-879.[3] R.C. S. Chandel and H. Kumar, Determination of phase shift defference for binomial

potential function, J~n–an–abha, 28 (1999), 67-76.

46

[4] A Erdélyi et al., Higher Transcendental Functions, V0l.1 Mc Graw - Hill. New York,1953.

[5] H.Kumar, R.C.S. Chandel and R.D. Agrawal, Phase shift involving Srivastava andDaoust function, J~n–an–abha, 29 (1999), 117-127.

[ 6] H. Kumar and V.P. Singh, Approximation formulae for phase shifts of s-waveSchrödinger equation due to binomial potential and their applications, AfricanJournalof Mathematics and Computer Science Research (Accepted, for publication 2010).

[7] Y.L. Luke, Integrals of Bessel Functions, Mc Graw -Hill, New York, 1962.[8] G.B.Mahajan and R.C. Varma, Determination of phase shifts for Rydberg potential

function, Indian J. Pure and Applied Physics, 13 (1975), 816-819.[9] S.S.Raghuwanshi and L.K. Sharma, On phase shifts for a model potential, Indian J.

Pure Applied Physics, 17 (1979), 102-105.[10] E.D. Rainville, Special Functions, Macmillan, New York, 1960, Reprinted:Chelsea Publ.

Co. Bronx, New York, 1971.[11] H.M. Srivastava and H.L.Manocha, A Treatise onGenerating Functions, John Wiley

and Sons, 1984.[12] T. Tietz, A new method for finding the phase shifts for the Schrödinger equation,

Acta. Phys. Hung., 16 (1963), 289-292.

47


A SHORT NOTE ON EXTON'S RESULTBy

M. Kamarujjama and Waseem A. Khan

Department of Applied Mathematics, Z.H. College of Engineering and TechnologyAligarh Muslim University, Aligarh-202002, Uttar Pradesh, India

E-Mail : [email protected]; [email protected]

(Received : December 25, 2010)

ABSTRACTIn this paper, authors derived some generating functions (partly bilateral

and partly unilateral) involving exponential and Mittag-Leffer's functions in viewof Exton's result [3].2000 Mathematics Subject Classification : 33E99Keywords: Mittag-Leffler's function and related function, Hypergeometricfunction.

1. Introduction and Definition. The function

0 1

k

k

zE z ,

k 0 (1.1)

was introduced by Mittag-Leffler's [5] and was investigated systimatically byseveral other authors (for detail, see [2, Chapter XVIII]).

The function

k

,k

zE z ,

k

0, 0 (1.2)

has properties very similar to those of Mittag-Leffler's function E z (See Wiman

[9], Agarwal [1]).

In 1971, Prabhakar [7] introduced the function ,E z in the form

kk

,k

zE z ,

k k!

0

, , 0 (1.3)

where k is the Pochhammer symbol (Rainville [8])

k, ... k

01 1 2 1 .

The function ,E z is most natural generalization of the exponential

function exp(z), Mittag-Leffler function E z and Wiman's function ,E z .

We note that

48

, , , ,

z, ,

E z E z ,E z E z ,E z F , ; z

E z E z E z e ,E z cosh z

11 1 1 1

1 21 1 1 1 1 2

11

(1.4)

An interesting (partly bilateral and partly unilateral) generating function

for mnF x , due to Exton [3, p.147(3)] is recalled here in the following (modified)

from [see [6]):

m n mn

m n m*

xtexp s t s t F x ,

s

(1.5)

where m mn nF x F n;m ; x / m! n! L x / m n !, 1 1 1 (1.6)

and mnL x denotes the classical Laguerre polynomials, (see [8] and in what folllows

m* max , m , m Z , , ,... 0 0 1 2 (1.7)

so that all factorials in equation (1.5) have meaning.2. Generating Reltions.

Result-1. If p,q,lN, then

lrt n / qp q m n

m n m* r

xxtexp s t s t

s m lr n lrr ! ! !

p q

0 (2.1)

Special Cases(i) For p=q=l, equation (2.1) reduces to

1 1 1

p m npp p

m n m*

n / p ;xt s texp s t F x .

m/ p ;s m/ p ! n / p ! (2.2)

(ii) When p=2 in (2.2) or p=q=l=2 in equation (2.1), we have

22 2 2

1 1

22 12 2

m n

m n m*

n / ;xt s texp s t F x .

m/ ;s m/ ! n / ! (2.3)

(iii) When p=q=1 in equation (2.1), we get

1

1 1

1

1

1 11

l

m njl l

lm n m*

j

n j;

ls texp s t xt / s F x .

m! n! m j;

l

(2.4)

For l=1, equation (2.4) reduces to (1.5). It can also be obtained from (2.1) by

49

taking p=q=l=1.

Result-2. If p,q,l N and ,E is defined by (1.3), then

31 2

1 1 2 2 3 3

lp q, , ,E s E t E xt / s

1 2 3

1 2 3

01 2 3 1 2 3

lrm lr n lrn / qm n r

p q

m lr n lrm n m* r a rp q

xs t

. (2.5)

Special Cases.

(i) For 1 2 3 1 , equation (2.5) reduces to

1 1 2 2 e e

lp q

, , ,

xtE s E t E

s

1 2 3

m n

m n m*

s t

lrn / q

m lr n lrr rp q

x

1 2 30 1 2 3

. (2.6)

(ii) When p=q=l in equation (2.5), we get

1 1 2 2 3 3

pp q

, , ,

xtE s E t E

s

1 2 3

m n

m n m*

s t

1 2 30 1 2 3

prn / p

m nr r r rp p

x . (2.7)

For 1 2 3 1 2 3 1 equation (2.7) reduces to (2.2) and also to

(1.5) for p=1.

(iii) If 1 2 3 1 and p q l 1 in equation (2.5), we get known

generating function of Kamarujjama and Khursheed [4]

1 1 2 2 3 3, , ,

xtE s E t E

s

1 2 3

m n

m n m*

s t

rn

r m r n r r

x

1 2 30 1 2 3

. (2.8)

(iv) When p=2 in equation (2.7) or p=q=l=2 in equation (2.6), we get

1 1 2 2 3 3

22 2

, , ,

xtE s E t E

s

1 2 3

m n

m n m*

s t

e

rn /

m nr r r r

x

1 2

22

0 1 2 32 2

. (2.9)

50

For 1 2 3 1 2 3 1 equation (2.9) reduces to (2.3).Now putting , 1 2 3 1 2 32 1 in equation (2.9) and using a

relation (1.4), we obtain a generating function of hypergeometric function 2F3 interms of hyperbolic cosine functions.

2 2

22 3

1 24

1 1 2 1 22 2

m n

m n m*

n, n / ;s tcoshs cosht cosh xt / s F x /

m ,m / , / ;m ! n !

. (2.10)

(v) For p q 1 , equation (2.6) reduces to

1 1 2 2 3 3

l

, , ,

xtE s E t E

s

1 2 3

m n

m n m*

s t

lrn / l

r m lr n lr r

x

1 2 30 1 2 3

. (2.11)

For 1 2 3 1 2 3 1 , equation (2.11) reuces to (2.4) and also to

(1.5) for l=1.

(vi) For 1 2 3 1 and l=1, equation (2.11) reduces to a following

generating relation

1 1 1 1 1 2 1 1 3

1 2

1 1 1m n

m n m* m n

s tF , ;s . F , ;t . F , ; xt / s

, n ;. F x

m, ;

2

2 21 3

1 1(2.12)

REFERENCES[1] R.P. Agarwal, A product d'ume note de Mpierre Humbert, C.R. Acad. Sci. Paris, 236

(1953), 2031-2032.[2] A.Erdélyi et al., Higher Transcendental Functions, Vol.3, McGraw-Hill, New York,

1953.[3] H. Exton, A new generating function for associated Laguerre polynomials and resulting

expansions, Jnanabha , 13 (1983), 147-149.[4] M. Kamarujjama and M. Khursheed Alam, Some generating relations involving Mittag-

Leffler's function, Proc. International Conf. SSFA (India), 2 (2001), 15-20.[5] G.M. Mittag-Leffler, Surla representation analyique d'une branche uniforme d'une

function monogene, Acta Math., 29 (1905), 101-182.[6] M.A. Pathan and Yasmeen, A note on a new generating relation for a generalized

hypergeometric function, J. Aust. Math. Soc., (1) 22 (1988), 1-7.[7] T.R. Prabhakar, A sigular integral equation with a generalized Mittag-Leffler function

in the kernal, Yokohana Math. J., 19 (1971), 7-15.[8] E.D. Rainville, Special Functions,The Macmillan Co, New York, 1960, Reprinted: Chelsea

Publ. Co., Bronx, New York, 1971.[9] A. Wiman, Uber den fundamentalsatz in der teorie der funktionen

xE , Acta Math.,

29 (1905), 191-207 and 217-234.

51


APPLICATIONS OF SAIGO FRACTIONAL CALCULUS OPERATORSFOR CERTAIN SUBCLASS OF MULTIVALENT FUNCTIONS WITH

NEGATIVE COEFFICIENTSBy

B.B. Jaimini

Department of MathematicsGovernment Girls Postgraduate College, Jhalwar-326001, Rajasthan, India


Jyoti Gupta

Department of MathematicsGovernment College, Kota-324001, Rajasthan, India

E-Mail:[email protected]

(Received : April 20, 2011)

ABSTRACT

In the present paper, we introduce the subclass jS n, p, ,q, of functions

with negative coefficients, which are analytic and multivalent in the unit disc

U z : z 1 . The fractional calculus of functions associated with ingegral

operator c ,pJ in the class jS n, p, q, as applications of the Saigo fractional

calculus operator , ,,zI

0 are established here.

Corresponding to our main theorems some known and unkown results forthe multivalent functions are also shown to be deduced as the special cases.2000 Mathematics Subject Classification : 30C45Keywords: Saigo Fractional Calculus Operators, Subclass of multivalent functions

1. Introduction. Let T j, p be the class of function of the form

p kk

k j p

f z z a z

ka ; p, j N , ,... , 0 1 2 (1)

which are analytic and p-valent in the open disc U z : z 1 . A function

f z T j, p is said to be p-valently close to convex of order in U if it satisfies

the inequality

52

pRe z f z 1 z U ; p; p N 0 . (2)

A function f z T j, p is said to be p-valently starlike of order in U if

it satisfies the inequality

zf zRe

f z

z U ; p; p N 0 . (3)

Further more a function f z T j, p is said to p-valently convex of order in U

if it satisfies the inequality

zf zRe

f z

1 z U ; p; p N 0 . (4)

For each f z T j, p we have [2]

q p q k q

kk j p

p! k!f z z a z

p q ! k q !

q N N ; p q 0 0 . (5)

For a function f(z) in class T(j,p), we define the following differential operator:

q qp,D f z f z 0 ,

q q q

p,

zD f z Df z z f z

p q

11

p q k qk

k j p

k qp! k!z a z

p q ! p q k q !

,

q q q

p, p, p,

zD f z D D f z z D f z

p q

12 1 1

2p q k q

kk j p

p! k! k qz a z

p q ! k q ! p q ,

and so on

q q qn n n

p, p, p,

zD f z D D f z z D f z

p q

11 1

53

n

p q k qk

k j p

p! k! k qz a z

p q ! k q ! p q

p, j N ;q N ; p q; 0 0 (6)

A function qf z T j, p is said to be in jS n, p, ,q, if and only if

qnp,

qnp,

z D f z 'Re ;

D f z (7)

where z U , , p N ,q,n N , p q, p q 00 0 and np,D is defined in (6).

We note that for q 0 the operator np,D f z in view of multiplier transformation

is defined and studied recently by Agharaly et al. [1] and Singh et al. [13] for positivecoefficients in the following form

n

p kp k

k p

kI n, f z z a z

p

1 ,n Z 0 .

Earlier the operator I n,1 was investigated by Cho and Srivastava [6] and Cho

and Kim [7]. Whereas the operator I n,l1 was studied by Uralegaddi and

Somanatha [16]. I n,1 0 is the well known Salagean derivative operator [11]

defined as

n n kk

k

D f z z k a z ,n N NU

02

0 .

2. Coefficient Estimate.

Theorem 1. Let the function f(z) defined by (1) be in the class T j, p .

Then the function f(z) belongs to the class jS n, p, ,q, if and only if

n

kk j p

k qk q k,q a p q p,q

p q

(8)

p q; p, j N ;q,n N ; p q; ; 00 0

where

54

01 101

,qp p ... p qp!p,q

,qp q ! (9)

Proof. Let us suppose the inequality (8) holds true. Then in view of (7), we have

nk p

qn kp, k j p

nqnk pp,

kk j p

k qk p k,q a zz D f z ' p q

p qD f z k q

p,q k,q a zp q

n

kk j p

n

kk j p

k qk p k,q a

p qp q

k qp,q k,q a

p q

Therefore the values of function

qn

p,

qnp,

z D f zz

D f z (10)

lie in a circle which is centered at w p q and whose radius is p q . Hence

the function f(z) satisfies the condition given in (7)

Now conversely, assume that the function f(z) is in the class jS n, p, ,q, . Then

we have

nk q

qn kp, k j p

nqnp, k q

kk j p

k qp q p,q k q k,q a zz D f z p q

Re ReD f z k q

p,q k,q a zp q

(11)

for some p q; p, j N ;q,n N ; p q; 00 0 and z U . Choose value of

z on the real axis so that z given by (10) is real. Upon clearing the denominator

in (11) and letting z 1 through the real values we can see that

n

k j p

k qp q p,q k q k,q

p q

55

n

k kk j p

k qa p,q k,q a

p q

, (12)

which leads to inequality (8). It completes the proof of Theorem 1.

Corollary 1. Let the function f(z) defined by (1) be in the class jS n, p, ,q,

then the following inequality hold true

k n

p q p,qa

k qk q k,q

p q

(13)

k j p; p, j N ;q,n N ; ; p q 0 0 .

The result is sharp for the function f(z) given by

p kn

p q p,qf z z z

k qk q k,q

p q

(14)

k j p; p, j N ;q,n N ; ; p q 0 0 .

3. Applicatrions of Fractional Calculus. In our present investigation,

we shall make use of the familiar integral operator c ,pJ defined by [5,p.676, eq.

(1.8)]

z c

c,p c

c pJ f z t f t dt

z

1

0 f T j, p ;c p; p N . (15)

Definition 1. The Reimann-Liouville fractional integral of order is defined, for

a function f z , by [15,p.224,eq.(3.1)]

z

z

fD f z d

z

10

1, (16)

where 0, f z is an analytic function in a simply-connected region of the z-

plane containing the origin, and hte multiplicity of z 1 is removed by requiring

log z to be real when z 0 .

Definition 2. The Reimann-Liouville fractional derivative of order is defined,

56

for a function f(z), by [15,p.224, eq. (3.2)]

z

z

fdD f z d

dz z

0

11 , (17)

where , f z 0 1 is an analytic function in a simply-connected region of the z-

plane containing the origin, and the multiplicity of z 1 is removed as in

Definition 1 above.Definition 3. Under the hypotheses of Definition 2, the fractional derivative of

order n is defined by [15,p.225, eq. (3.3)]

n

nz zn

dD f z D f z

dz , (18)

where and n N NU 0 0 .

Srivastava, Saigo and Owa defined the following fractional integral operatorinvolving Gauss's hypergeometric function:

Definition-4. For real number , 0 and , the Saigo fractional integral

operator , ,,zI

0 is defined by [12,p.112, Eq. (8)] (See also [15]):

z, ,

,z

zI f z z F , ; ; f d

z

10 0

1 , (19)

where f(z) is an analytic function in a simply-connected region of the z-planecontaining the origin with the order

f z O z , as z 0 ,

where max , 0 1 and the many-valuedness of z 1 is removed by

requiring log z to be real when z 0 .

From Definition (1) and Definition (4), it is easy to see that

0, ,

z ,zD f z I f z .

Lemma. If 0 and k 1 , then [12]

, , k k,z

k kI z z

k k

0

1 11 1 . (20)

57

For a function f z T j, p defined by (1), we obtain easily the following results

in view of (15) and (20) respectively :

p kc,p k

k j p

c pJ f z z a z

c k

c p; p, j N (21)

and

0

1 1 1 11 1 1 1

, , p k,z k

k j p

p p k kI f z z a z

p p k k .(22)

Now for the function c ,pJ f z defined in (21) we obtain the following fractional

integral in view of (22)

, , p,z c ,p

p pI J f z z

p ( p )

0

1 11 1 –

kk

k j p

k kc pa z

c k k k

1 11 1 (23)

,c p; p, j N 0 .

Theorem 2. Let , and satisfy inequalities , p , p , 0 1 1

p 1 . Choose a positive integer such that 1n / p .

If jf z S n, p, ,q, then

, ,,z c ,p

p p c pI J f z

p p c p j

0

1 11 1

j p

n

p j p j p q p,qz z

j p qp j p j j p q j p,q

p q

1 1

1 1

0 00 0z U ; p q;c p; p, j N ;q,n N ; p q; (24)

and

, ,,z c ,p

p p c pI J f z

p p c p j

0

1 11 1

58

j p

n

p j p j p q p,qz z

j p qp j p j j p q j p,q

p q

1 1

1 1

0 00 0z U ; p q;c p; p, j N ;q,n N ; p q; , (25)

where

U, p

UU , p

0 0 .

These results are sharp for function f z given by

p j pc,p n

c p p q p,qJ f z z z .

j p qc p j j p q j p,q

p q

(26)

Proof. To prove the Theorem 2, we start from eq. (23), i.e.

, , p,z c ,p

p pI J f z z

p p

0

1 11 1

kk

k j p

k kc pa z

c k k k

1 11 1 . (27)

Now on setting

, ,

,z c ,p

p pH z z I J f z

p p

0

1 11 1 (28)

the above result given in (27) takes the following form

p kk

k j p

H z z a k z

(29)

where

k k p p c pk

k k p p c k

1 1 1 11 1 1 1 .

Since k is a decreasing function of k, therefore

59

k p j

1 1 1 1

1 1 1 1

j p j p p p

j p j p p p

c p

c p j

(30)

Now in view of Theorem 1, we have

k nk j p

p q p,qa .

j p qj p,q j p q

p q(31)

Hence for the function H z obtained in (29), we have

p j pk

k j p

H z z z j p a

.

Therefore, making an appeal to (30) and (31), we obtain

p c p j p j p p pH z z

c p j j p j p p p

1 1 1 11 1 1 1

j p

n

p q p,qz .

j p qj p q j p,q

p q

Now in view of (28), we atonce arrive at the desired result in (24).The second result of Theorem 2 is proved similarly in view of (30) and (31) forinequality

p j pk

k j p

H z z z j p a

.

4. Special Cases. (I) If in differential operator qnp,D f z defined in (6) we

consider 0 then it reduces to the operator studied by Aouf [2]. Therefore, at

0 all the results established in the Sections-2, 3 will provide the known results

of Aouf [2].

(II) The results obtained in class jS n, p, ,q, here in turn provide many

known results studied in various subclasses by specializing the parameters

j,n, p,q, , . To illustrate we give few classes as follows:

60

(i) 0j jS n, p, ,q, S n, p,q, (Aouf [2]) ,

(ii) j jS , p, ,q, S p,q, 0 0 and j jS , p, ,q, C p,q, 1 0 (Chen et al.[4]),

(iii) jS n, , , , P j, ,n j N ;n N ; 01 0 0 0 1 (Aouf and Srivastava

[3]),

(iv) 1 01 0 0 0 1S n, , , , T n, n N ; (Hur and Oh [8]),

(v)

*j

j

T p, OwaS , p, , ,

T p, j Yamakawa

100 0 0

17 ,

(vi)

*j

j

C p, OwaS , p, , ,

CT p, j Yamakawa

101 0 0

17 ,

(vii) S , p, , , T * p, 1 0 0 0 and S , p, , , C p, p N ; p 1 1 0 0 0

(Owa [9] and Salagean et al. [11]),

(viii) jS , , , , T j 0 1 0 0 and 01 1 0 0 0 1jS , , , , C j n N ;

(Srivastava et al. [14]),

(ix) j jS n, p, , , S n, p, p, j N ,n N ; p 00 0 0 (Aouf [2]),

(III) If in (24) and (25) we take then these inequalities reduce to the

following inequalities involving R-L fractional integral operator defined in (16).

Corollary 2. Let the function f z defined by (1) be in the class jS n, p, ,q, .

Then

z c,p

pD J f z

p

11

j p

n

c p p j p q p,qz z

j p qc p j p j j p q j p,q

p q

1

1

(32)

z U ; p q; ;c p; p, j N ;n N ; p q; 00 0 0 ,

and

61

z c,p

pD J f z

p

11

j p

n

c p p j p q p,qz z

j p qc p j p j j p q j p,q

p q

1

1(33)

z U ; p q; ;c p; p, j N ;n N ; p q; 00 0 0 .

Each of the assertion (32) and (33) is sharp for the function f(z) givey by (26).

The results in (32) and (33) in turn at 0 give the known results due to Aouf [2,

p.32, eq. (6.6) and eq. (6.7)].REFERENCES

[1] R. Aghalary, Rosihan M.Ali, S.B. Joshi and V. Ravichandran, Inequalities for analyticfunctions defined by certain linear operators, Int. J. Math. Sci., 4 No.2 (2005), 267-274.

[2] M.K. Aouf, On certain multivalent functions with negative coefficients defined by usinga differential operator, Mat. Vesnik, 62 No.1 (2010), 23.35.

[3] M.K. Aouf and H.M. Srivastava, Some families of starlike functions with negativecoefficients, J.Math.Anal. Appl. 203 No.3 (1996), 762-790.

[4] M.P. Chen, H. Irmak and H.M. Srivastava, Some multivalent functions with negativecoefficients defined by using differential operator, Pan Amer. Math.J. 6 (1996), 55-64.

[5] M.P. Chen, H. Irmak and H.M. Srivastava, Some families of multivalently analyticfunctions with negative coefficients. J. Math. Anal. Appl., 214 No. 2 (1997), 674-690.

[6] N.E. Cho and H. M. Srivastava, Argument estimates of certain analytic functionsdefined by a class of multiplier transformations. Math. Comput. Modelling, 37 No. 1-2 (2003), 39-49.

[7] N.E. Cho and T.H. Kim, Multiplier transformations and strongly close-to-convexfunctions, Bull. Korean Math. Soc. 40 No.3 (2003), 399-410.

[8] M.D. Hur, and G.H. Oh, On certain class of analytic functions with negative coefficients,Pusan Kyongnam Math. J. 5 (1989), 69-80.

[9] S. Owa, On certain classes of p valent functions with negative coefficients, SimonStevin 59 No.4 (1985), 385-402.

[10] S. Owa, The quasi-Hadamard products of certain analytic functions, in :H.M.Srivastava, S. Owa (Eds.) Current Topics in Analytic Function Theory, World ScientificPublishing Company, Singapore, New Jersey, London and Hong Kong, 1992, 234-251.

[11] G.S. Salagean, Subclasses of Univalent Functions, Lecture Notes in Math. (Springer-Verlag) 1013 (1983), 362-372.

[12] T. Sekine, Distortion theorem for fractional integral operator, Univalent functionsand the Briot-Bouquet differential equations (Japanese) (Kyoto, 1996).

62

Su–rikaisekikenkyu–sho Ko–kyu–roku No.963 (1996), 110-11813.[13] S. Singh, S. Gupta and S. Singh, On starlikeness and convexity of analytic functions

satisfying a differential inequality. J. Inequal. Pure Appl. Math., 9 No.3 (2008), 14.[14] H.M. Srivastava, S. Owa and S.K. Chatterjea, A note on certain classes of starlike

functions, Rend. Sem. Mat. Univ. Padova 77 (1987), 115-124.[15] H.M. Srivastava and S. Owa (Eds.) Univalent Functions, Fractional Calculus and

Their Applications, Halsted Press (Ellis Horwood Limited, Chichester), John Wileyand Sons, New York, Chichester, Brisbane, Toronto, 1989.

[16] B.A. Uralegaddi and C. Somanatha, Certain Classes of Univalent Functions, in CurrentTopics in Analytic Function Theory, H.M. Srivastava and S. Owa (ed.), World Scientific,Singapore, (1992), 371-374.

[17] R. Yamakawa, Certain subclasses of p valently starlike functions with negativecoefficients, Current Topics in Analytic Function Theory, 393-402, World Sci. Publ.River Edge, NJ, (1992).

63


SOME RESULTS ON A -GENERALIZED RIEMANN ZETA FUNCTIONBy

R.K. Gupta and Mina Kumari

Department of Mathematics and StatisticsJai Narain Vyas University, Jodhpur-342005, Rajasthan, India

E-Mail:[email protected]; [email protected]

(Received : October 05, 2010; Revised: October 08, 2011)

ABSTRACTIn this paper we give a -generalization of the new zeta function due to

Goyal and Laddha. This function is defined by series and the corresponding integralrepresentations are established. We have also investigated the properties of thenew type of generating functions such as integral representations and generatingfunctions. Several interesting results obtained earlier by Goyal and Laddha [3],Katsurada [4] and Bin-Saad [1] follow special cases of our main findings.2010 Mathematics Subject Classification : 11M35, 11S23.Keywords: generalized zeta function, integral representation, bionomialtheorem, and gamma function.

1. Introduction. An interesting definition of the zeta functions, due toGoyal and Laddha [3], is as follows:

*

0

; , , 1,!

nn

zn

yy z a y

a n n (1.1)

Re 0, 1,a where

n

n.

This is expressed as the integral form

* 1

0

1, , 1z at ty z a t e ye dt

z . (1.2)

This function is continued to a meromorphic function over the whole z-plane (cf.Erdélyi [2]). Obviously, when =1, (1.1) reduces to the zeta function studied byErdélyi [2]. Katsurada [4] introduced two hypergeometric type generatingfunctions of the Riemann zeta function as follows:

0

,!

m

zm

xe x z m

m x , (1.3)

64

and

0

; ,!

m

z mm

xf v x v z m

m 1x (1.4)

where v and z are arbitrary fixed complex parameters.Recently, Bin-Saad [1] has defined two new type of generating functions suggestedby (1.3) and (1.4) as :

0

, ; , ; , ,!

m

m

xx y z a y z m a

m 1y (1.5)

0

, ; , ; , ,!

m

v mm

xx y z a v y z m a

m 1,y x a (1.6)

where is the generalized zeta function.

Its various properties are investigated including the integralrepresentations, generating functions, partial sums and N-fractional calculus. Ina recent paper the authors [5] have defined a new generalized zeta function andderived its hypergeometric types of generating functions. In this paper we aim atgiving -generalizations of the generalized zeta function and at deriving theirvarious properties and formulas including their integral representations, seriesand genrating functions.

2. Definition and Integral Representations. A -generalization ofgeneralized zeta function is defined in terms of series representations as :

*

0

; , ,!

nn

zn

yy z a

a n n , (2.1)

where 1,Re 0, 1, , 0.y a t R t

It is interesting to note that for 1 , (2.1) reduces to the generalized zeta function

studied by Goyal and Laddha [3].Here we prove the following integral representation

* 1

0

1; , , 1 ,z at ty z a t e ye dt

z (2.2)

where 1,Re 0, 1, , 0.y a t R t

Proof of (2.2). Let

1

0

11z at tI t e ye dt

z

65

1

00

1!

ntnz at

n

yet e dt

z n

On interchanging the order of integration and summation, we see that

1

00

1

!

np zn

zn

yI e p dp

zn a n

0 !

nn

zn

y

a n n

* ; , ,y z a .

This completes the proof of (2.2).We now give the -generalization of two hypergeometric type generating functionsof the Riemann zeta function in the form:

0

; ,!

m

zm

xe x z m

m (2.3)

, , 0,x R

and

0

; , ,!

m

z mm

xf v x v z m

m (2.4)

1, , 0,x R t

where v and z are arbitrary fixed complex parameters.

3. Integral Involving ;ze x and ; ,zf v x . In this section we evaluate

definite integrals involving the function ;ze x and ; ,zf v x in terms of other

kind of zeta and hypergeometric functions. First, we recall the Eulerian integralformula of first kind (cf. Srivastava and Manocha [6]);

1 11

01 ,Re 0,Re 0.yx x y

t t dt x yx y (3.1)

From the term by term integration, we can derive the following

Theorem 1. Let Re 0c b and Re 0b . Then

66

1 11

01 ; , ; ,c bb

z z

ct t e xt dt G b c x

c b b (3.2)

and

1 11

01 ; , , , ; ,c bb

z z

ct t f v xt dt G v b c x

c b b , (3.3)

where

0

, ; ; , ,!

mm

zm

b mc xG b c x z m

b c m m (3.4)

and

0

, ; , ,!

m

zm

c b m xG b c x z m

b c m m (3.5)

For 1 , (3.2) and (3.3) reduce to the known results given earlier by Katsurada

[4].Proof of (3.2). Denote for convenience the left-hand side of relation (3.2) by I.Then in view of (2.3), it is easily seen that

1 11

00

1!

mc bb m

m

cxI z m t t dt

m c b b .

Using (3.1), we obtain

0 !

m

m

c b m c bxI z m

m c b b c m

0

.!

m

m

c b m xz m

b c m m

After some simplification, we get the right-hand side of (3.2).Proof of (3.3). Denote for convenience the left-hand side of relation (3.3) by J.Then in view of (2.3), it is easily seen that

1 11

00

1!

mc bb mm

m

v x cJ z m t t dt

m c b b .

Upon using the Eulerian integral formula (3.1) and the definition (3.4), we arefinally led to right-hand side of formula (3.3).

67

Theorem 2. Let Re 0z , Re 0 and Re 1, , 0R Then

1 1

0 0

1, ; , ,z av vu v e xue y z a dudv

z

*

0

; , ,n

zn

x yz a

a n a n, (3.6)

and

1

0

1, ; , ,z at t

vt e xe y z a dtz

*

0

; , ,n

v zn

x yz a

a n a n, (3.7)

where

0

, ; , , ; , ,!

m

m

xx y z a y z m a

m (3.8)

1, , 0,Re 1, 0, 1, 2,...y R z a

and

0

, ; , , ; , ,!

m

v mm

xx y z a v y z m a

m (3.9)

1, , 0,Re 1, 0, 1, 2,...y R z a ,

while is the -generalized zeta function.

For =1, (3.8) and (3.9) reduce to the hypergeometric type generating functionstudied by Bin-Saad [1].Proof of (3.6). Denote for convenience the left-hand side of equation (3.6) by L.Then, in view of (3.8), we have

1 1

0 0, 0

1 1

!

m nv a mm z

z mm n

x yL u e du e v dv

zm a n .

Applying the Eulerian integral for gamma function, we get

0 0 !

mn

mz z

n m

x a nyL

a n m a m.

68

After some simplification, we obtain the right-hand side of (3.6).This completes the proof of (3.6). Proof of (3.7) can be developed on the

same line.ACKNOWLEDGEMENT

We are grateful to Prof. R.K. Saxena, for his constant encouragement inpreparing this paper, and to the referee for his suggestions to bring the paper in itspresent form.

REFERENCES[1] M.G. Bin-Saad, Sums and partial sums of double power series associated with the

generalized zeta function and their N-fractional calculus, Math. J. Ohayama Univ., 49(2007), 37-52.

[2] A. Erdélyi et al., Higher Transcendental Functions, Vol. I, McGraw-Hill, New York,1953.

[3] S.P. Goyal and R.K. Laddha, On the generalized Riemann zeta-functions and thegeneralized Lambert transform, Ganita Sandesh, 2 (1977), 99-108.

[4] M. Katsurada, On Mellin-Barners type of integrals and sums associated with the Riemannzeta-functions, Publicastions de L'Institut Mathematique, Nouvelle Series, Tome 62 (76)(1997), 13-25.

[5] R.K. Saxena, R.K. Gupta and Mina Kumari, Integrals and series expansions of the -generalized zeta function, Indian Academy Mathmatics, 33 No.1, 2011 (to appear)

[6] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press,Brisbane, London, New York (1984).

69


SOLUTION OF QUADRATIC DIOPHANTINE EQUATIONSBy

Pratap Singh

Department of Mathematics and StatisticsSam Higginbottom Institute of Agriculture, Technology and Sciences,

(Deemed-to-be-University) Allahabad-211007, Uttar Pradesh, India

(Received : July 10, 2011)

ABSTRACTOur aim is to solve the quadratic Diophantine equations

1161146329226x2–y2= 1. Two starting least positive integer solutions are

obtained.2000 Mathematics Subject Classification : Primary 11GXX, 14GXX;Secondary 11D09.Keywords: Quadratic Diophantine equation, Quadratic irrational number, Periodof Continued fraction, Convergent of continued fraction.

1. Introduction. In Diophantine equations only positive integer solutions

are calculated. Solutions of quadratic Diophantine 2 19 7nx and 2 11 3nx

are discussed by Devi [1]. Unique integer solution of the equation 2 13 3nx is

discussed by Sharma, Singh and Harikishan [3]. Ternary cubic Diophantineequation is solved. Different patterns of non-zero positive integer solutions areobtained by Gopalan and Somnath [2]. Our aim is to find the positive integersolution of quadratic (in two variables) Diophantine equations.

2. Formation of the problem. Consider the quadratic Diophantine

equations 2 2 1Nx y , where N is a positive integer. Let 0N x be the quadratic

irrational number. The algorithm for N generates the infinite simple continued

fraction of the form 0 1 2 1, , ,..., , ,...n nq q q q q as follows

0 0 10 0

1 1 21 1

1

1,

1,

... ....1

,n n nn n

q x xx q

q x xx q

q x xx q

70

We do not have to calculate infinitely many iq s . Since the quadratic

irrational number is always periodic. So it is of the form

0 1 2 1, , ,..., , ,...,n n n mq q q q q q . It is also found that 02n mq q . Number of terms m,

from qn+1 to qn+m is called period of continued fraction. If the period is odd boththe equations have positive integer solution.

We denote the nth convergent /n n nC P Q , therefore

0 00

0

0 111

1 1

1 2

1 2

;1

1;

, 2n n n nn

n n n n

P qC

Q

q qPC

Q q

P q P PC for n

Q q Q Q

(2.2)

The solutions of the equations 2 2 1Nx y are

1 , ix Q y P for 0,2,4,...i and

,i ix Q y P for 1,3,5,...i . respectively.

3. Results and Discussion.

1161146329226 1077565,2155130 .

In this case, period of continued fraction is one, which is odd. Therefore both theequations can be solved.From (2.1) and (2.2), we get

0 0 1 1 21077565, 1, 2322292658451, 2155130, 5004842577008581195,P Q P Q P

2 3 34644585316901, 10786086382990825883438801, 100096885154015007260Q P Q

Solutions of the equations 1161146329226x2–y2= 1 are

x y1 10775654644585316901 5004842577008581195 andx y2155130 2322292658451100096885154015007260 10786086382990825883438801

Only two starting least positive integer solutions are calculated.

71

4. Conclusion. There will be infinite number of solutions. In this problemonly two solutions of each of the equation are calculated. One may find the solution

of other Diophantine equation of the form 2 2 ,Nx y n where n is a positive integer.

REFERENCES[1] Usha Devi, On the Diophantine euqation x2+19=7n and x2+11=3n. The Mathematics

Education, 39, 2 (2005), 84-86.[2] M.A. Gopalan, Manju Somnath, Ternary Cubic Diophantine equation

2 1 2 2 32 x y z , Acta Ciencia Indica, 34, 3 (2008), 1135-1137.

[3] P.K. Sharma, M.P. Singh and Harikishan, On the Diophantine euqation 2 13 3nx .

Acta Ciencia Indica, 34, 3 (2008), 1043-1044.

73


q-OPERATIONAL FORMULAE FOR A CLASS OF q-POLYNOMIALSUNIFYING THE GENERALIZED q-HERMITE AND q-LAGUERRE

POLYNOMIALSBy

R.K. Kumbhat, R.K. Gupta and Madhu Chauhan

Department of Mathematics and StatisticsJ.N.V. University, Jodhpur-342005, India

E-Mail : [email protected], [email protected]

(Received : October 15, 2010; Revised : March 20, 2011)

ABSTRACTIn the present paper, certain q-operational formulae for the generalized q-

polynomials ; , ,nJ x r p l are developed and make an attempt to unify the various

results. The results given earlier by Gould and Hopper [6], Singh and Srivastava[9], Al-Salam [1], Das [4], Carlitz [3], Joshi and Singhal [10] follow as special cases.2000 Mathematics Subject Classification : Primary 33D05; Secondary 33D45,33D50.Keywords: q-operational formulae, q-Hermite and q-Laguerre polynomials,Fractional q-derivative, q-Gamma function.

1. Introduction. Burchnall [2] made use of the opertional formula

0

2 1 ,n

n n k kn k

k

nD x H x D

k / ,D d dx ...(1.1)

to prove the well-known relation

min ,

0

2 !m n

km n m k n k

k

m nH x K H x H x

k k . ...(1.2)

Gould and Hopper [6] studied operational formulae associated with classicalpolynomials and established that

0

1 , , ,n

n kn r kn k

k

nH x r p D

k

D ...(1.3)

where the symbol D is defined by1 /rD prx x D

and satisfies the relation

1

0

,n

n n r

j

x D xD prx j

74

and

, , 1 expnr r n rnH x p x exp px D x px ...(1.4)

defines the elegant generalization of the Hermite polynomials to which it reduces

when 0, 1, 2.p r

The relation (1.3) provides a generalization of the formula of Burchnall [2]quoted above as well as of Carlitz's formula [3]

1 0

! ,!

knnk k

n kj k

xxD x j n L x D

k ...(1.5)

for the Laguerre polynomials.Joshi and Singhal [10] also introduced certain operational formulae associatedwith a class of polynomials unifying the generalized Hermite and Laguerrepolynomials.The Rodrigues formula (c.f. Joshi and Singhal [10]) is

ln; , , , exp exp ,r n rnJ x r p l c l n x px D x px ...(1.6)

where

1 2 / 2

1 / 22

1, ,

2 1

n l l

nl lnl l

c l n

l being a non-negative integer.In a recent paper, the authors [8] established certain operational formulae

for the generalized basic Laguerre polynomials with the help of known results.The object of the present paper is to define q-analogue of the generalized polynomials

; , ,nJ x r p l and develop certain q-operational formulae for the generalized

polynomials ; , , ,nJ x r p l q and make an attempt to unify the various results.

For real or complex ,0 1q , the q-shifted factorial is defined as

1 1

1 , 0; ;

1 1 ... 1 ,nn n

na q q q

q q q n N . ...(1.7)

In terms of the q-gamma function, (1.7) can be expressed as

1; , 0

nq

nq

n qq n , ...(1.8)

where .q is the q-gamma function (c.f. Gasper and Rahman [5]) given by

75

1

;

; 1q

q q

q q q .

Indeed, it is easy to verify that

1lim qq and

1

;lim ,

1nn nq

q q

q

...(1.10)

where

1 ... 1 , 1n

n n . ...(1.11)

The fractional q-derivative of arbitrary order 0 for a function

1f x x , is given by

11

,q

q xq

xD x , ...(1.12)

where 0, 1, 2,... .

For 1 , the equation (1.12) reduces to

1 221

,

1

1 1q

q xq

q xxD x

L q

. ...(1.13)

Further, we shall denote the infinite product

1 1 2

0 1 21

1 ... 1 , ,..., ;

, ,..., ;1 ...

j j

r r

j jj ss

a q a q a a aq

b b bb q a b q ...(1.14)

In what follows the other notations and symbols employed in this paperhave their usual meanings.

2. q-Extension of ; , , .nJ x r p l In this section, we define a q-extension of the

generalized polynomials ; , ,nJ x r p l due to Joshi and Singhal [10], by means of

the following relation

1 2 / 2

ln,1 / 2

2

1; , , , ,

2 ;

n l l rq n r

n q x qn l l

nl l

x e pxJ x r p l q D x e px

l q

...(2.1)

76

where r,p,l and constants assume integral values, and the q-Leibnitz rule is

, , ,0

nn n r rq x q x q x

r

nD UV D U D V

r

, ...(2.2)

where

;

; ;n

r n r

q qn

r q q q q

. ...(2.3)

By virtue of (1.12) and (1.10), we observe

2

1lim 1 ; , , ,

n l

nq

q J x r p l q

; , ,nJ x r p l . ...(2.4)

3. The q-Operational Formulae. In this section, we prove the various q-

operational formulae by working use of the q-differential operator qxD , which

possesses the following interesting properties:

(i) ; ;F x f x q x F f x q , ...(3.1)

(ii) ;qF e g x f x q ' ;qe g x F xg f x q , ...(3.2)

(iii) ... 1nx F F F n nx F . ...(3.3)

We now consider the expression

ln ln r n r

q q qe px x D x e px Y

ln ln r n n n r

q q qe px x x D x e px Y

1 2 ... 1 .ln ln r n r

q qe px x n x e px Y [By (3.3)]

1

1n

n r

j

x l n prx j

[By making an appeal to (3.1) and (3.2)].

We thus have

ln n rq qD x e px Y

1

1ln

n

n r rq

j

x e px l n prx j Y .

On the other hand, replacing n by n-k, by lk in (2.1), we find that

77

1 / 22nl ln ln r a rq q qD x e px Y x e px

1 2 / 2

20

1 1;n

n k l l

n k l lk

nq

k

1 /2

; , , ,2

kllk

n kl l

xj x r p l q . ...(3.5)

Comparision of (3.4) and (3.5), gives the q-operational formula

1 2 / 21 1 / 2

1 0

1 2 1nn

n k l ll n nl lr

j k

nl n prx j x

k

2 / 22

1; ; , , ,2

kllk

n kl ln k l l

xq J x r p l q . ...(3.6)

Also

r n a ln rq q qx e px D x e px Y

11 2 ... 1 l n kn r n k r

q qx e px n x x e px Y .

Therefore, making an appeal to (3.1), (3.2), (3.3) and (3.5), we obtain the another q-operational formula

11 n l n k r

qx k x e px Y

1 2 / 21 / 2

0

2 1n

n k l lnl lk n rq

k

nx e px

k 1 / 221;

2

kl

l ln k l l

xq

; , , , .lk kn k qJ x r p l q D Y ...(3.7)

Next we observe that the q-analogue of the recurrence relation of Joshi andSinghal [10]:

1 1 ; , , ,l l l rq nx D x prx J x r p l q

1

1

,; , , ,

, 1 n

c l nJ x r p l q

c l n

suggests the q-operational formula

, ; , , ,mq l nJ x r p l qD

,

; , , ,,

nlm n

c l nJ x r p l q

c l m n ...(3.8)

where1 1

,l l l r

q L qx D x prx D ,

78

which corresponds to the q-analogue of the formula of Gould and Hopper [6] towhich it reduces when l=0.

4. Particular Cases. In this section we shall deduce some interestingparticular cases of the q-operational formulae.

(i) If we let –1q and make use of the limit formula (1.10), in (3.8), we obtain

known results due to Gould and Hopper [6].(ii) If we put l=0 and make use of (1.10), our formula (3.6) reduces to (1.3) due to

Gould and Hopper [6].(iii) Again, if we set p=l=r=1 in (3.6) and make use of the limit formula (1.10), we

arrive at the known results due to Carlitz [3].

(iv) On the other hand, when l=1, –1q and use of (1.10), (3.7) yields the known

formula due to Joshi [7].(v) If we put l=1, p=r=1 k=0, Y=1 in (3.7), we get the q-operational formula of

Das [4]:

1 ; ;n

n nq q q nn

xD x e x x e x q q L x q .

(vi) If we take l=k=0, Y=1 in (3.7), we obtain

1 1 ; , ,n nn r n r

q nx xDq x e px x H x r p q .

5. Applications. Setting Y=1 in (3.6), we have

(i)

1

1 1n

r

j

l n prx j

1

; , , ,,

l n

n

xJ x r p l q

c l n , ...(5.1)

so that

1

; , , ,,

l m n

m n

xJ x r p l q

c l m n

1

1 1m n

r

j

l m n prx j

1

1m

r

j

l m n prx j n

1

1 1n

r

j

l m n prx j

1

1

1

1 . ; , , ,,

l n ml nr

nj

xn l m prx j J x r p l q

c l n .

Therefore in views of (3.6) we finally derive

79

(ii)

, ,; , , ,

, m n

c l m c l nJ x r p l q

c l m n

1 2 / 2

02

11

1;

mk l l

kml l

mkq

1 / 22

1;2

kl

l lm k l l

xq

1; , , , ; , , ,l mlk n km k q nJ x r p l q D J x r p l q . ...(5.2)

If we reverse the order of the operator on the left hand side in (3.6) andproceed as above, we obtain an alternative q-operational formula

(iii)

, ,; , , ,

, m n

c l m c l nJ x r p l q

c l m n

1 2 / 2

02

11

1;

mk l l

kml l

mkq

1 / 22

1;2

kl

l lm k l l

xq

; , , , ; , , , .lk lmk

m k q nJ x r p l q D J x r p l q ...(5.3)

Comparision of (5.2) and (5.3) gives the identity

(iv)

1 2 /2

1 /220

1 1;2

klmk l l

l lm k l lk

m xq

k

1; , , , ; , , ,n lk l mkm k q nJ x r p l q D J x r p l q

1 2 / 2

1 /220

1 1;2

klmk l l

l lm k l lk

m xq

k

; , , , ; , , ,lk k lm

m k q nJ x r p l q D J x r p l q , ...(5.4)

which in the special case l=0, making use of (4.6), gives

(v)

0

1 ; , , ; , ,m

k r k rm k q n

k

mH x p q D H x r p q

k

0

1 ; , , ; , ,m

k r k r mm k q n

k

mH x n p q D H x q p q

k ...(5.5)

for the generalized q-Hermite polynomials.

Next in (5.3) if we replace by lm , multiplying on both the sides by

80

;

m

m

tq q and summing from m=0 to m=, we arrive at

(vi)

2

0

;; , , ,

;m n l l m lm

m nm m

l qt J x r p l q

q q

2

20

;1; , , , , ; , , ,

;ml l lmr m

n l mnl lm m

l qq J x A tx r p l q t J x r p l q

q q

where

1 2 /2 1 /21 2k l l l llA ...(5.6)

ACKNOWLEDGEMENTWe are grateful to Prof. R.K. Saxena, for his constant encouragement in

preparing this paper. We are also thankful to the referee for his suggestions tobring the paper in its present form.

REFERENCES[1] W.A. Al-Salam, Operational representations for the Laguerre and other polynomials,

Duke Math. J., 31 (1964), 127-142.[2] J.l. Burchnall, A note on the polynomials of Hermite, Quart. J. Math Oxford, 12 (1941),

9-11.[3] L. Carlitz, A note on the Laguerre polynomails, Mich. Maths J., 7 (1960), 219-223.[4] M.K. Das, Operational representation for the Laguerre polynomials, Acta Math. Acad.

Sci. Hungar. 18 (1967), 335-338.[5] G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press,

Cambridge, New York (1990).[6] H.W. Gould and A.T. Hopper, Operational formulae connected with two generalizations

of Hermite polynomails, Duke Math. J., 29 (1962), 51-63.[7] C.M. Joshi, Operational formulae associated with the generalized Laguerre polynomials,

Mathematica, 11 (34), 2, (1969), 269-276.[8] R.K. Kumbhat, R.K. Gupta and Madhu Chouhan, Certain operational formulae associated

with generalized basic Laguerre polynomials, Vijanana Priadhad Anushandan Patrika52, (2), (April 2009), 163-172.

[9] R.P. Singh and K.N. Srivastava, A note on generalization of Laguerre and HumbertPolymials, La Ricerca, (2), 14 (1963), 11-21. Errata, ibid (2)15 (1964), 63.

[10] J.P. Singhal and C.M. Joshi, On the Unification of generalized Hermite and Laguerrepolynomials, Revista Matemática Hispanoamericana, ISSN 0373-0999, 42, 1-3 (1982)82-89.

81


ON DISCRETE HERMITE TRANSFORM OF GENERALIZEDFUNCTIONS

ByS.R. Mahto

Department of MathematicsGossener College, Ranchi-834001, Jharkhand, India


ABSTRACTIn this paper discrete Hermite transformation of generalized functions

belonging to a certain testing function space has been defined and an inversionformula has been established.2010 Mathematics Subject Classification : 44A20, 46F10Keywords: Hermite Transformation, Generalized Functions.

1. Introduction. The aim of the present work is to extend the Hermitetransform

2xnF n H f x e H x f x dx

(1.1)

and its inversion formula

0

12 ! nn

n

F nf x H x x

n (1.2)

studied by Debnath [1], to a class of generalized functions. Zemanian [3,4] has alsostudied (1.1), but here the present work is treated differently.

2. The Notation and Terminology. Throughout this work the interval(-,) is denoted by I. D(I) denotes the space of infinitely differentiable functionson I which have compact support in I. The topology of D(I) is that which makes itsdual D'(I) the space of Schwartz distributions. E(I) denotes the space of all infinitelydifferentiable functions on I. Its dual E'(I) is the space of distributions with compact

support. The symobls kx denotes the operator

2 2 k

k x xx x xD e D e 2 2 2

k

x xD xD , /xD d dx (2.1)

3. Testing Function Space H(I) and Its Dual H'(I). We define H(I) asthe space of all complex valued infinitely differentiable functions (x) on I suchthat

sup kk x

xx

, 0,1,2,3,...k . (3.1)

It is seen that H(I) is a linear space. The topology of H(I) is that generated by the

82

countable multinorm , 0,1,2,3,....k k .

H'(I) is the dual of H(I) and is equipped with the usual weak topology. The membersof H'(I) are called generalized functions. It can be seen than H(I) is a completecountably multinormed space.We now give below some properties of the space H(I) and it dual H'(I).

(i) D I H I and topology of D(I) is stronger than the topology induced on it

by H I . Hence, the restriction of any member of 'H I to D I is in 'D I .

(ii) D I H I E I . As D I is dense in E I . H I is also a dense subspace

of E I . Consequently, 'E I can be identified as a subspace of H'(I).

(iii) For each 'f H I , there exists a non-negative integar r and a positive

constant C such that for all H I .

0

, max kk r

f C (3.2)

The proof of this statement follows from the boundedness property ofgeneralized functions.

(iv) If f(x) be a function of x defined in the interval ( , ) such that

f x dx

, then f(x) generates a regular generalized function of H'(I)

defined by

, , .f f x x dx H I (3.3)

This result can be easily established as follows

0, .f f x dx

(v) For each positive integer n, the function 2

,xne H x x is a member

of H(I). This can be proved as follows :

2 2

2x xx n ne H x n e H x

Hence,

2x

x ne H x 2

sup k xx n

xe H x

2

sup 2 k xn

xn e H x

, 0,1,2,3,...x k .(3.4)

83

4. The Discrete Hermite Transform of Generalized Functions.Members of H'(I) are called Discrete Hermite transformable generalized functions.

The Discrete Hermite transform of 'f H I is defined as an application

of 'f H I to the kernel 2xne H x H I ,

i.e. 2

, , 0,1,2,... .xnF n H f f x e H x n (4.1)

As we require time to write the expression

0

12 !

Nn n

nn

H x H tn

,

therefore again, we denote it by , ;nT t x N being any positive integer and

,x .t We now state below several lemmas whose proofs are based upon the similar lemmasproved in Dube [2].

Lemma 4.1. Let ' ,f H I then for any positive integer N and for any arbitrary

x D I

2 2

, ,R t x

NRf t e T t x e x dx

2 2

, ,R t x


(4.2)

Lemma 4.2. 2

lim , 1R x

NRNT t x e dx

.

Lemma 4.3. Let x be an arbitrary member of D(I) with compact support in

(–R,R) I, then

2

, 0R x

NRT t x x t e dx

as N .

5. Inversion Theorem. We now prove the following inversion theoremfor our Discrete Hermite transform:

Theorem 5.1. If F(n) denotes the Discrete Hermite transform of 'f t H I as

defined in (4.1), then in the sense of convergence in 'D I ,

0

1lim

2 !nnN

n

H xf t F n

n

.

Proof. Let x D I be an arbitrary member of D(I). We are to prove that

0

1,

2 !

Nnn

n

H xF n x

n

, f t t as N . (5.1)

84

Now 2

, ,xx D I e x D I hence (5.1) is equivalent prove

2

0

1,

2 !

Nn xn

n

H xF n e x

n 2

, xf t e n as N . (5.2)

We have

2

0

1,

2 !

Nn xn

n

H xF n e x

n

(5.3)

2

0

12 !

NR n xnR

n

H xF n e x dx

n

(5.4)

[As x D I , the support of x is contained in (–R,R)].

2 2

0

1,

2 !

NR n x xnnR

n

H xf t e H t e x dx

n

(5.5)

2 2

0

1,

2 !

NR n nt xnR

n

H x H tf t e e x dx

n (5.6)

2 2

, ,R t x


(5.7)

2 2

, ,Rt x


(5.8)

2

, tf t e t asN . (5.9)

(5.3) equals to (5.4) as the function

0

12 !

Nnn

n

H xF n

n is locally integrable over the

interval I and x is in D(I) with support in (–R,R).

(5.6) is obvious because of linearity property of functionals. From Lemma 4.1 weobtain (5.8). Finally, Lemma (4.3) helps us to derive (5.9).

REFERENCES[1] L. Debnath, On Hermite transform, Maths. Vest., 1 (16) (1964), 285-292.[2] L.S. Dube, On finite Hankel transformation of generalized functions, Pac. J. Math.,

62(2) (1976), 365-378.[3] A.H. Zemanian, Orthonormal series expansion of certain distributions and distributional

transform calculus, J. Math. Anal. and Appl. 14 (1966), 263-275.[4] A.H. Zemanian, Generalized Integral Transformations, Inter Science Publishers 1968.

85


DRS–TRIANGULATION BASED ON 2N–RAY ALGORITHMBy

R.S. Patel, Ankit Agarwal and Lakshmi Narayan Tripathi

Department of Mathematices, Satna Postgraduate College, Satna, M.P.E-mail:[email protected], [email protected]

(Received : November 05, 2010; Revised: March 20, 2011)

ABSTRACTIn order to improve the efficiency of the 2N-Ray algorithm; we propose a

variant of the DRS-Triangulation. A nice property of this triangulation is that itsubdivides all the subsets, on which the 2N-Ray algorithm works, into simplicesaccording to the DRS-Triangulation. Numerical tests shows that 2N-Ray algorithmbased on D1/2-Triangulation is much more efficient.2010 Mathematics Subject Classification : 55M20, 54H25; Secondary 57Q15Keywords: Artificial algorithm, triangulation, admissible lebelling, nodes,barycentre, extra layer.

1. Introduction. Various triangulations have been situdied by Dang [1],Dang and Talman [2], Scarf [4], Todd [6,7], Vander and Talman [8], Vertgeim [9].The 2n-ray algorithm was proposed by Wright in [10] to compute solutions of non-linear equations. The 2n-ray algorithm partitions Rn into 2n cones which have thesame vertex. Then a triangulation of Rn subdivides each cone into simplices. The2n-ray algorithm starts at the vertex and leaves it along an edge of some cone. Itfollows a sequence of adjacent simplices with varying dimension. Under some mildconditions, the 2n-ray algorithm terminates at an n-dimensional simplex that yieldsan approximate solutions to the system of non-linear equations. Since these 2ncones have 2n edges, each of which is a ray, the 2n–ray algorithm has 2n possibleray to leave the vertex.

Motivated by this work, we have tried to develop a new DRS–triangulation.The DRS-triangulation based on the 2n ray algorithm is much more efficient than2n ray algorithm proposed by Wright [10] as well as K1 triangulation introducedby Kuhn in [3] and J1 triangulation given by Todd [5].

2. Notations and Definitions. The following notations have been usedin this paper :R: Set of real numbers,Z: Set of all integers,N: Set of positive integers (1,2,...,n),

N0: Set of all non-negative inegers 0N ,

86

Rn : n dimensionally space, having co-ordinates indexed 1 through n,Rn+1: n+1 dimensional space, with coordinates indexed 0 through n,: Group of permutation on (1,2,...,n) and +1 group of permutation on

(0,1,2,...,n),

ui : ith unit vector in Rn, jN and i

i Nu u

,

mR : Non negative orthant of . . ; 0m mR i e x R x .

Now we consider some standard definitions and explanations which will beused in this paper.

2.1 Standard Simplex. The standard n dimensional closed simplex Sn is

the convex hull of 0 1 1, ,..., . . : 1n n n Tv v v i e S x R v x . n

is denotes the face of sn

opposite . . : 0i n ni iv i e S x s x and boundary of ns is denoted by n

i N is U s .

Again a j-dimensional simplex or [j-simplex] is the relative interior of the

convex hull of j+1 affinely independent points 0 1 2 3, , , ,..., jy y y y y , called its vertices.

We write 0 1 2, , ,..., jy y y y . A simplex is a face of if its vertices are a subset

vertices of the . It is convemient to call the closure of a (j–1) dimensional face ofthe j simplex as a facet of . Two j simplices are said to be adjacent if they sharea common facet.

2.2 Triangulation. A trangulation G of Sn is a collection of n simplicesand satisfies the following two conditions :1. The simplices in G together with all their faces form a partition of Sn and2. Each point of Sn has a neighbourhood meeting only a finite number ofsimplices.2(a) Pivot Rule. For a given simplices G and a vertex y of the rules for obtainingthe simplex of G whose vertices include all vertices of except y, are called thepivot rules of G.

2(b) Mesh. The mesh of a tringulation G is sup G diam . We shall use the

Euclidian norm through out this paper.

2.3 Definition. For each sign vector ns R , let

E(s)={ :ni ix R s x x whenever 0is )

= cone { :nt R t is a sign vector, and 0i i is s t } .

In case s has k non-zero components for k>0 than E(s) is a polyhedral cone of

dimension 1n k . Also we have 0 nE R . Moreover, when 0s each nE s B

87

is a polyhedral of a cubical subdivisions of nB where nB donote the unit ball in

1 norm.Wright [10] has also defined another subdivision of Rn into closed convex

cone as n-dimensional geometric form given as follows :

0 if 0

:0 if 0

i in

i i i

x sLetC s x R

s x s

=cone {siui :si0, for each sign vector s}. If s has k non-zero componentsthen C(s) is an orthant of a k-dimensional coordinate subspace of Rn.

Wright [10] has proposed two type of T-triangulations of Rn with the property

that E(t) is a subcomplex for every sign vector nt R for 0t . The first

triangulation is called a K1 triangulation. This triangulation is obtained by takingthe triangulation K1 due to Kuhn [3] in the first orhant and the reflecting throughcoordinate hyperplane to triangulate the other orhant. A vector v1 of a n-simplex<v1,...,vn+1> of K1 specified by choosing a sign vector s with all the non-zerocomponents, is a member of C(s) when all its components are integrals and is apermutation of {1,2,...,n}, then v1K1 is defined recursively as :

1 ,ii i

iV V S U for i=1,2,...,n.

For n=2 triangulation K1 can be illustrated by following diagram.

The second triangulation J1 is defined by Todd [5] and which can beillustrated for n=2 by the following figure :

2.3.1. The DRS-Triangulation. Define

1: max , 2,3,...,n niW x R x x i n

taking a vector 1 2, ,...,T

nY y y y , we have

X2

X1

(central vertices are heavy dots)

88

if is even1 otherwise,

i ii

i

x xY

x

where [] is the greatest integer less than or equal to . Let D is the set of allnY W where iY is defined above. If Y D , we define

1; iI y i N y y and 1:T

jJ y j N y y .

Let 1 2, ,...,T

ns s s s be a sign vector such that

1. For i N , if 0iy then 1,is and if 0iy then 1is .

Let , : 1iK y s i I y s .

Let denote the number of element in I(y) and h the number of elements in

,K y s , we take integer p such that

1. when h=0, if =n then p=0 or 2,

2. when h>0, if h=n then p=0 and if h<n then 0 1p n .

Let 1 , 2 ,..., n be permutation of N.

When h=0, for j=1,2,...,n,1. If j=1, define

1 if 0 otherwisei

i I yg j

...(1)

for i=1,2,...,n.

2. If 1j , we define

if ,0 otherwise

i

i

S i jg j

...(2)

for i=1,2,...,n.When h>0, for j=1,2,...,n.

1. If , ,j K y s define

11 if , and 0 otherwisei

i K y s j ig j

...(3)

for i=1,2,...,n,

89

2. If ,j K y s , define

if 0 otherwise

ni

s j i jg j

...(4)

for i=1,2,...,n.

If y, , s and p be as above, then vectors 0 1, ,..., ny y y are defined as follows :

for p=0, we have0y y

, 1,2,...,ky y g k k n ...(5)

and for p1, we define0 ,y y s

1 , 1,2,..., 1.kk k

ky y s U k p

, ,..., .ky y g k k p n ...(6)

The 0 1, ,..., ny y y vectors obtained from the above definition are affinely independent.

Thus their convex hull is a simplex. Let us denote this simplex by , , ,RSD y s p or

0 1, ,..., ny y y . Let RSD be the set of all such simplices. Then DRS is a triangulation

of Wn. Note that simplices of the DRS-triangulation can be represented in morethan one way. Moreover, the triangulation of a whole cube in Wn is the same as theD1-triangulation.

To be more expatiate let us illustrate DRS-triangulation of Wn for n=2 and

for x4. Obviously, we have that for 1 4y .

0,0,0 , 2,2,0 , 4,4,0 , 4,0,4 , 4,4,4T T T T TD . ...(7)

1. Let 0,0,0 Ty . Then, 1: 1,2,3iI y i N y y and =3.

Then s must be (1,1,1)T. Thus

1, : 1 1,2,3K y s i I y s and 3h . We have 0p .

(a) Let 2,3,1 . Then 1 3,1,2 and by applying (3)

1 2 31 2 2 , 2 , 2g g g g g

90

1,1,1 T ,

2 3 1,0,1 Tg g ,

3 3 1,0,0 Tg g .

Therefore, 0 0,0,0 ,Ty y

1 1 1,1,1 ,Ty y g

2 2 1,0,1 ,Ty y g

3 3 1,0,0 ,Ty y g Let 1 0 1 2 3, , , .y y y y

(b) Let 3,2,1 . Then 1 3,2,1

1 3 1,1,1 ,Tg g

2 2 1,1,0 ,Tg g

3 1 1,0,0 ,Tg g

Therefore, 0 0,0,0 ,Ty y

1 1 1,1,1 ,Ty y g

2 2 1,1,0 ,Ty y g

3 3 1,0,0 ,Ty y g Let 2 0 1 2 3, , ,y y y y .

2. Let 2,2,0 Ty . Since 1: 1,2iI y i N y y and =2.

So s must be (-1,-1,1)T. Thus , ; 1ik y s i I y s and h=0 while p can be any

one of 0,1,2,.Now by applying (1) and (2), we have

1 1, 1,0g

2 0, 1,0g

3 0,0,1g .

91

Now considering different values of p and applying (6) and (7), we obtain thefollowing simplices:

(a) For p=0. Let 1,2,3 . Therefore,

0

1

2

2,2,0 ,

1 1,1,0 ,

2 2,1,0 ,

T

T

T

y y

y y g

y y g

3 3 2,2,1 Ty y g . Let 3 0 1 2 3, , ,y y y y .

(b) For p=1. Let 1,2,3 . Therefore

0

1

2

3 4 0 1 2 3

1,1,1 ,

1 1,1,0 ,

2 2,1,0 ,

3 2,2,1 . Let , , , .

T

T

T

T

y y s

y y g

y y g

y y g y y y y

(c) For p=2. Let 1,2,3 . We have,

0

11 01

2

3 5 0 1 2 3

1,1,1 ,

2,1,1 ,

2 2,1,0 ,

3 2,2,1 . Let , , , .

T

T

T

T

y y s

y y s u

y y g

y y g y y y y

3. Let 4,4,0 Ty . Therefore, 1; 1,2iI y i N y y and 2 .

We have that s must be (-1,-1,1)T. Thus , : 1iK y s i I y s and h=0.

We have that p can be any one of 0,1,2. We also haveg(1) = (-1,-1,0),g(2) = (0,-1,0),g(3) = (0,0,1).

(a) For p=0, and 1,2,3 , we have

0

1

2

3 6 0 1 2 3

4,4,0 ,

1 3,3,0 ,

2 4,3,0 ,

3 4,4,1 . Let , , , .

T

T

T

T

y y

y y g

y y g

y y g y y y y

92

(b) For p=1, let 1,2,3 . Therefore

0

1

2

3 7 0 1 2 3

3,3,1 ,

1 3,3,0 ,

2 4,3,0 ,

3 4,4,1 . Let , , , .

T

T

T

T

y y s

y y g

y y g

y y g y y y y

(c) For p=2 and 1,2,3 , we have

0

11 01

2

3 8 0 1 2 3

3,3,1 ,

4,4,1 ,

2 4,3,0 ,

3 4,4,1 . Let , , , .

T

T

T

T

y y s

y y s u

y y g

y y g y y y y

4. Let 4,0,4 Ty . Since 1; 1,3iI y i N y y and 2 . so that s must

be (-1,-1,1)T. Thus , : 1iK y s i I y s and h=0. we have that p can be any

one of 0,1,2. We also haveg(1) = (-1,0,-1),g(2) = (0,1,0),g(3) = (0,0,-1).

(a) For p=0, and 1,2,3 , we have

0

1

2

3 9 0 1 2 3

4,0,4 ,

1 3,0,3 ,

2 4,1,4 ,

3 4,0,3 . Let , , ,

T

T

T

T

y y

y y g

y y g

y y g y y y y


0

1

2

3 10 0 1 2 3

3,1,3 ,

1 3,0,3 ,

2 4,1,4 ,

3 4,0,3 . Let , , ,

T

T

T

T

y y s

y y g

y y g

y y g y y y y

(c) For p=2 and 1,2,3 , we have

93

0

11 01

2

3 11 0 1 2 3

3,1,3 ,

4,1,3 ,

2 4,1,4 ,

3 4,0,3 . Let , , ,

T

T

T

T

y y s

y y s u

y y g

y y g y y y y

5. Let 4,4,4 Ty . Since 1; 1,2,3iI y i N y y and 3 . We have that

a must be (–1,–1,–1)T. Thus ,K y s and h=0. we have that p can be any one of

0,1,2. We also haveg(1) = (–1,–1,–1),g(2) = (0,–1,0),g(3) = (0,0,–1).

(a) For p=0. Let 1,2,3 . Therefore

0

1

2

3 12 0 1 2 3

4,4,4 ,

1 3,3,3 ,

2 4,3,4 ,

3 4,4,3 . Let , , , .

T

T

T

T

y y

y y g

y y g

y y g y y y y


0

1

2

3 13 0 1 2 3

3,3,3 ,

1 3,3,3 ,

2 4,3,4 ,

3 4,4,3 . Let , , , .

T

T

T

T

y y s

y y g

y y g

y y g y y y y

(c) For p=2. Let 1,2,3 . We have,

0

11 01

2

3 14 0 1 2 3

3,3,3 ,

4,3,3 ,

2 4,3,4 ,

3 4,4,3 . Let , , ,

T

T

T

T

y y s

y y s u

y y g

y y g y y y y

It can be seen that : 1,2,...,14i i form a triangulation of W3 for 4x

as shown by the following figure :

94

REFERENCES[1] C. Dang, The Di–Triangulation of Rn for simplicial algorithm for computing solutions

of nonlinear equations, Mathematics Operations Research, 16 (1989), 148-161.[2] C. Dang and A.J.J. Talman, The (2n+1–2)-ray algorithm based on a variant of the Di-

triangulation to compute stationary points on the unit simplex, Center Discussion paperNo. 9050, Tilburg University, The Netherlands (1990).

[3] H.W. Kuhn, Simplicial approximation of fixed points, Proc. Nat. Acad. Society, U.S.A.,61 (1968), 1238-1242.

[4] H. Scarf, On the Computational of Equilibrium Points. Ten Essays in Honour of IrvingFisher, edited dy W. Fellner Wiley, New York, (17) (1967), 207-230.

[5] M.J. Todd, The Computation Fixed Point and Application, Lecture Notes in Economicsand Mathematical System, 124, Springer Verlag New York (1976).

[6] M.J. Todd, Improving the convergence of fixed point algorithms, MathemticalProagramming Study, (7) (1978), 151-169.

[7] M.J. Todd, On triangulation for computing fixed points, Maths Programming, 10 (1976),322-346.

[8] G. Vander Lann and A.J.J. Talman, A restart algorithm for computing fixed points withan extra dimension, Math. Programming, 17 (1979), 74-84.

[9] B.A. Vertgiem, On an Approximate Determination of the Fixed Points of ContinuousMappings, Soviet Math. Dokl. 11 (1970), 295-298.

[10] A.H. Wrirght, The octahedral algorithm, A new simplicial fixed point algorithm, MathsProgramming, 10 (1981), 47-69.

95


COMPARISON OF A.D.D. OF TRIANGULATIONSBy

R.S. Patel, Ankit Agarwal and Preeti Singh Bhahel

Department of Mathematices, Satna Postgraduate College, Satna, M.P.E-mail:rssumit1963@rediffmailcom, [email protected]

(Received : December 12, 2010)

ABSTRACTIn this paper, making an appeal to computer we try to provide the measure

of efficiency for triangulations through graphical represenation.2010 Mathematics Subject Classification : 55M20, 54H25; Secondary 57Q15Keywords: A.D.D., Triangulations, Fixed Points, Complementary pivotingtechniques, Leveling of vertices of the triangulation, Homotopy, Simplex, Sandwich,Finer meshes, Directional density, Theoretical measures of efficiency.

1. Introduction. There are number of algorithms for comuting fixed pointsusing triangulations and complementary pivoting techniques. Each algorithmrequires a triangulation, a lebeling of the vertices of the trinagulation, and aparticular starting point called as pivoting point.

The successful algorithms use one of the two methods to obtain better andbetter approximations of fixed points, without wasting previous approximations.

The first one is the restart method of Merril [4], independently developedby Kuhn and Mackinnon [3]. called "sandwich" method. If a fixed point of a mapping

f on Rn is sought, one triangulates. 0,1nR rather than Rn. All vertices lie in

either 0nR or 1nR . The latter are labelled according to the given mapping

f, the former according to the constant map taking each x into nC R . The point c

can be so chosen as to be closed to a previous fixed point and provides a startingpoint for the algorithm.

The second method was introduced by Eaves [1] and extended by Eaves and

Saigal [2]. Instead of triangulating Sn or Rn, one triangulates 1,nS

or 1,nR . Roughly, triangulations of finer and finer meshes are placed on the

top of the another so that boundary simplices match. Then one generates an infinitepath of simplices following fixed point of piecewise linear maps that approximatesf better and better as the artificial coordinate tends to . This process ofdeformation corresponds to the "homotopy" concept. Since we are not concerned

96

with the maping f, we shall say such triangulations have continuous refinement ofgrid gize.

The efficiency of fixed point algorithm is very sensitive to the triangulationused. In this paper we are concerned with comparison of the theoretical measuresof efficiency for different types of triangulations. The technique of the comparisonis based on the concept of directional density of the triangulation in the directiond as defined by Todd [8]. The only other measures known are rather crude measurebased on the number of simplices in the unit cube and the diameter, introduced bySaigal, Solow and Wolsey see [7].

In this paper, we have also tried to provide the measure of efficiency forsuch triangulations through graphical representations by using the computer.

2. Notations and Definitions. The following notations have been usedin this paper :R: Set of real numbersZ: Set of all integersE : Set of all even integers.O: Set of all odd integersN: Set of all positive integers {1,2,...}.

N0: Set of all non-negative inegers 0N

Rn : n dimensional space, having co-ordinates indexed 1 through n.Rn+1: n+1 dimensional space, with coordinates indexed 0 through n.: Group of permutation on (1,2,...,n) and n+1 group of permutation on

(0,1,2,...,n)

ui : ith unit vector in Rn, iN and

i

i Nu u is a vector of ones in Rn+1.

Vj: jth unit vector in Rn+1, iN0 and j

j Nv v is a vector of ones in Rn+i.

mR : Non negative orthant of mR i.e. : 0mx R x .

A: Set of 1, 1

B : Set of 0,1 .

Now we consider some standard definations and explanations which willbe used in this paper.

2.1 Simplex. An (open) j-dimensional simplex 0 1 2, , ,..., jy y y y is the

relative interior of the convex hull of j+1 affinely independent points 0 1, ,..., jy y y

called its vertices. A simplex is a face of if its vertices are a subset of vertices of

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the . Again the closure of a (j–1)-dimensional face of the j-simplex is called afacet of . Two j-simplices are said to be adjacent if they share a common facet.

The diameter of is sup : , max : 0 .i jx z x z y y i j

2.2. Triangulation. A triangulation G of an m-dimensional subset F ofRn is a collection of m-simplices that satisfies the following two conditions :1. The simplices in G, together with all their faces form a partition of F; and2. Each point of F has a neighbourhood meeting only a finite number ofsimplices.

We denote Gj the set of j-dimensional faces of simplices in G for 0 j m .

The members of G0 is called the vertices of G-and the closures of members of Gm-

1 are called facets of G.

2.3. Pivot Rule. For a given simplex G and a vertex y of , the rules for

obtaining the simplex of G whose vertices include all vertices of except y arecalled the pivot rules of G.

2.4. Mesh. The mesh of a triangulation G is sup{diameter of : G }. We

shall use the euclidean norm throughout this paper.2.5 Some Standard Regular Traingulations. We define some regular

triangulations of Rn. We take the basic grid size of all triangulations to be unity,K,H and J1 are triangulations of Rn. K (some times called K1) is based on thestandard subdivision of the cube. H is very closely related to K and J1 is the "UnionJack" triangulation of Rn.

All these triangulations use integer vectors as vertices, let0 0 0

1nK H J Z . Also denote the set of central vertices of J1 by 0

1cJ ={ 0

1 : iy J y

is odd for each i N }.

Now we give definitions of some standard trianglations given as below :

(1) K triangulation is the set of 0 ,..., ny y where 0 1; ,ii iy y y y u 1 i n

(2) H triangulation is the set of 0 ,..., ny y where 0 ; ,'ii i iy y y y q 1 i n

where qj is the j-th column of the n×n matrix

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The simplex o is the set all nx R satisfying :

1

1 11 1

1 ... 0n

n nx y x y

.

(3) J1 trinagulation is the set of 0 ,..., ny y where 0 1; ,ii i

iy y y y s u

1 i n and 01, ,ocy j is a permutation of N and ns R is a sign vector, where is i .

The simplex is the-set of all ns R satisfying :

1 1 11 ... 0n n ns x Y s x Y .

The triangulations by J1 and by K of any unit cube with integral verticesare isomorphic upto orientation of the edges of the cube.

K2(m) and J2(m) are the triangulation of Sn having mesh 12K m m

and 12 2mesh J m m

. Each K2(m) and J2(m) can be expressed as follows :

(4) 02 2: , , if nK m k y S is the collection of all such where

0 ,..., ny y satisfying 1 1 ii iy y m q for each i N and where qj is the jth

column of the (n+1)×n matrix denoted by Q and expressed as

and its vertices are given by 02 :n

iK m y S my Z for each 0i N .

(5) 02 2: , , , if nJ m j y s S is the collection of all such where

0 ,..., ny y satisfying 1 1 ii i

iy y m s q for each i N and iq is as defined

in (4) and its vertices are given by 0 02 2J m K m and centre vertices

0 02 2 :c

iJ m y J m my is even for 1 i n and nmy is odd} for each 0i N

We now describe two triangulation with continuous refinement of grid size.The first J3, based on J1 was introduced by Todd [8], and K3 is due to Eaves and

99

Saigal [2].

(6) K3 triangulates (0,1]×Rn. Let 03K denotes set of vertices {y(0,1] ×Rn:y0

=2–k for 0 k Z and 0/iy y z for i N }. A mapping 0 13: c nt K A is defined by

ti(y)=0 or if yi/y0 is odd and 1 otherwise. (Thus t0(y) always equals 0). Let be a

permutation of N0 and K3 be the set of 1 0, , ,..., nk y y y y where

1

1 10

1 00

1

10

,0 0

, ,

2 .

ii i

n ni kj j

ij k j

kk k

y y

y y y v i j

y y y t v v v

y y y v j k n

The simplex is the set of all 1nx R satisfying

0 0 0 0 00 0 0 ... .n ny x y t x y x Y t x y y x

(7) J3 triangulates (0,1]×Rn. Let 03J denotes set of vertices { 1

0: 2n ky R y

for 0 K Z and 0/iy y z for 1 i n }.

Let 0 03 3 0: / for 1c

iJ y J y y O i n be the set of central vertices. A

mapping 0 13: c nJ A is defined by 1i y or +1 according as yi/y0 is 1 or 3

mod 4. Let is a permutation of N0 and ns A . Then J3 is the set of simplex

3 , ,j y s 1 0, ,..., ny y y where

1

1 10

10

10

,

,0 0 ,

,

2 ,1 .

ii ii

nj j

j

kk k

y y

y y y s v i j

y y y v

y y y k v k n

The simplex is the set of all 1nx R satisfying :

0 0 01 1 1 10 1 ... j j jy s x y s x y x y

0 01 1 1 ...j j J n n nx y x y y x .

(8) 3K triangulates (0,1]×Rn. The set of vertices of 3K , denoted by 03K is

precisely 03J . We define 0 1

3: nK B by

100

01 / mod2.i iy y y

Let n is a permutation of N0 with 0j and 0y if 1 j then

3K is the set of 1 0, , ,..., nk y y y y where

1

10

11 0

0 0 00 1

10

,0

;

2 .

ii i

j ni kj j

ci k j

kk k

y y

y y y v i j

y y y v y v y v

y y y v j k nLet z=x–y, =(y) and w=z+z0. Then simplex is the set of all xRn+1

satisfying :

0 00 ... .ny w w w

(9) Let A be a non-singular n×n matrix and G be any triangulation of Rn (herewe consider G=K or J1). Let AG= (AG} with A={Ax:x}. Then trinagulationAK of Rn is the collection of simplices =(y0,...,yn) such that all components ofA-1y0 are integers and yi=yi-1+a(i) where a(j) is the jth column of A. When A has1's on the diagonal, -1's on the upper diagonal, and zeroes elsewhere, we have the Htriangulation.(10) J' is a triangulation of Rn defined as follows : The set of vertices of J' is theset of vector vRn with each component an integer, such that there is not precisely

one even component or precisely one odd component. Each simplex ' , ,j y s

0 ,..., ny y , where y is the vector each of the whose components is an even integer,

1 ,..., n is a permutation of (1,2,...,n) and s is a sign vector (each 1js )

with 1 1ns s .

For convenience n i

is e is denoted by ie , where en(i) is the ith unit vector in

Rn. For n4 the vertices of j' are defined by

0

1 0 1

2 1 1 2

1

1 2 1

1

2 ,

,

,3 2,

,

2 .

j j i

n n n n

n n n

y y

y y e

y y e e

y y e j n

y y e e

y y e

101

For n=3, each simplices of J' is of the form 0 1 2 3' , , , , ,j y s y y y y where

0

11 0

2 32 12

33 2

,

2 ,

,

.

i

y y

y y e

y y e s e e

y y e

Now below we give three triangulations of K.1,J. and J.3 of 0,1nR given

by Todd [8].

Let 1 10,1 : 0n n nR x R x . Removal of the bar from a vector in

0,1nR denotes its projection into Rn. Let 1 1,..., nu u be the unit vector in Rn+1;

thus u1,...,un are the unit vectors in Rn.

(11) Suppose 0ny Z and is a permutation of {1,2,..,n+1}. Then 1 ,k y

denotes the (closed) simplex 0 1,..., ny y , where

0

1

,

,1 1.ii i

y y

y y u i n

The triangulation 1K is the set of all such 1 , 'k y s .

(12) Suppose 1ny Z has all iy 's odd. Let be a permutation of {1,2,...,n+1}

and let 1ns R be a sign vector, each is is 1. Then 1 , ,j y s denotes the

simplex 0 1,..., ny y , where

0

1 ,1 1ii ii

y y

y y s u i n

.

The triangulation 1J is the set of all such 1 , , 'j y s s .

(13) 3J is the triangulation derived from 3J . Let 1ny Z has all iy 's odd.

Let be a permutation of {1,2,...,n+1} with 1j n . Let 1ns R be a sign

vector such that, for 1,j k n sn k is -1 or +1 according as ky is 1 or 3 mod

4. Then 3 , ,J y s denotes the simplex 0 1,..., ny y , where

102

0

1

11 1

11

1

,1 .

,

2 , 1.

ii i

nkj j n

k j

kk k

y y

y y s i u i j

y y s k u s u

y y s k u j k n

The triangulation 3J is the set of all such 3 , ,J y s 's.

3. Measure of Efficiency for Triangulation. Todd [8] has indicated thatthe theoretical measure of the efficiency of different types of triangulations usedto be compared by counting the number of simplices into which unit cube is divided.However, Todd [8] has also observed that many triangulations have differentefficiency but yield the same number of simplices e.g. K, H or J1

triangulations

having n! simplices and K'3 or J3 in 1

0: 0 1,1/2 1nx R x x having

(2n+1–1) n! simplices.Saigal Solow and Wolsey introduced the concept of diameter, for those

triangulations that subdivided unit cube. This is the maximum over all pairs offacets of the triangulation lying in the facets of the cube as well as of the maximumnumber of simplices that form a path of simplices linking the two facets. Theycomputed the diameter of K triangulation and an obvious extension to K3

andfound them compareble, Saigal [5] calculated the diameters of triangulations Kand H and obtained results and suggesting that the number of iterations using Kincreases with n2 while that using H increases with n3. It is easy to see that thediameters of J1

and K are equal.The diameter of triangulation is a "worst best case" measure similar to the

diameter of a polytope which is of interest in linear programming.4. Main Result. First we shall give the concept of average directional

density, then formula for obtaining different triangulations and numerical resultsfor obtaining a.d.d. for different values of n. At last we shall give the computerisedgraph of average directional density of different triangulations.

4.1 Average Directional Density. Todd [8] proposed the concept ofdirectional density as an alternative measure, for the comparison of efficiecy ofthe triangulations and is global in nature.

For , nx d R and 0, let ,x x d denote : 0,x d . Let G be a

triangulation of Rn having mesh n . Let , , ,N G x d denote the number of

simplices of G intersecting ,x x d divided by . Let , , ,N G x d denote the

103

limit as , (if it exists) of the average of N{G,x,d,) for x uniformly distributed

in 0, :nB x R x p . Let N(G,d) be the limit as (if it exists) of

, ,N G d . Finally, N(G) is defined to the average of N(G,d) for d uniformly

distributed on : 1n nB d R d .

We call N(G,d) the directional density of G in direction d and N(G} theaverage directional density. It should be noted that N(G,x,d,1) is the number ofsimplices met per unit step size. To eliminate the effect of starting point, we averageover x lying in a large ball. To eliminate any effect on the ending point, we let

. Finally to get a measure independent of the direction d, we average over

nB .

For any triangulation G of Rn, Gn–2 is a countable collection of sets of

dimension n–2. Thus for almost all , :x x d R meets no simplices of G+ of

dimension less than n–1. If x and x+d lie in n-simplices of G.[x,x+d] meet onemore simplex of G than (n–1)-simplex of G. Thus the computation of N (G,x,,d)can be made by counting the number of points of [x,x+d] lying in facet of G, foralmost x.

From these observations the following results are obtained by Todd [8] andVender Laan and Talman [19] :

(i) 1 , i i ji i jN k d d d d

(ii) 1

1,

2i i j i ji i jN J d d d d d d

(iii)

1 , ki j i jN H d d

(iv) 1

',2 i j i ji j

N J d d d d d

(v) Let 2 / 2 / 1 1 / 2 ,ng n n n then

(a) 1 1 2 / 2 nN K N J n n g

(b) 1 11 nN H n i i g

(c) 1/ 2 1 1 .niN K n n i i g

(d) ' / 2 2 nN J n g

104

(e) 1 1 / 4 1 1k

N J n n k k

1/ 2 1/2

1

3 1 1 / 1 ni j n

i j i j n g

(f)

1/2

1/2

1 /8 2 if is even*

1 /8 +1 if is odd

n

n

g n n n n nN A K

g n n n n

Table 1 gives N(G)/gn for various values of a n and G equal to K, J1, H , H,

A*K and J'.TABLE 1

The a.d.d. of the K or J1, K , H, A*K, J 1 and J' triangulations for various

values of n(mesh equal to n )

TRIANGULATIONS G

n K or J1 K H A*K 1J J'

1 1 0.7 1 1 0.7 02 3.41421 2.44949 3.4142 2.44949 2.698 1.41423 7.24264 5.4641 7.5604 4.89898 6.4587 4.24264 12.4853 9.94936 13.7047 7.74597 12.098 8.48525 19.1421 16.0797 22.089 11.619 20 14.14219 59.9117 60.0249 82.0105 33.541 77.6782 50.911615 163.492 197.748 269.433 87.6356 261.087 148.492420 288.701 392.453 534.189 151.987 524 268.700630 645.183 1044.16 1419.95 334.066 1406 615.182950 1782.41 3636.58 4942.12 910.357 4336 1732.412100 7100.36 20108.5 27316.5 3588.52 27462 7000.357

4.2. Asymptotic Behaviour of A.D.D. of Different Triangulations. If

G is any triangulation then Asymptotic behaviour of /N G gn for 1, , ,G K J K

1,J H and AK are given as follows :(i) For K or J1 triangulation : 2

1/ ~ / 2 ' ,nN K g n N J N J

(ii) For K* triangulation : 5 / 2* / ~ /16,nN K g n

(iii) For H triangulation : 5 / 2/ ~ 4 15,nN H g n

(iv) For A*H triangulation : 2* / ~ / 8nN A K g n ,

(v) For 1J triangulation : 5/ 21 / ~ 0 .nN J g n

In table 2 we compared asymptotic behaviour of different triangulations

105

for different values of n.TABLE 2

The asymptotic behaviour of a.d.d. of the K or J1, K , H, A*K and J'

triangulations for various values of p (mesh equal to n )

Tringulations G

n K or J1 K H A*K 1J

1 0.70711 0.19643 0.26667 0.35355 12 2.82843 1.11117 1.50849 1.41421 5.656853 6.36396 3.06202 4.15692 3.18198 15.58854 11.3137 6.28571 8.53333 5.65685 325 17.6777 10.9807 14.9071 8.83883 55.90179 57.2756 47.7321 64.8 28.6378 24315 159.099 171.172 232.379 79.5495 871.42120 282.843 351.382 477.028 141.421 1788.8530 636.396 968.295 1314.53 318.198 4929.550 1767.77 3472.4 4714.05 883.883 17677.7100 7071.07 19642.9 26666.7 3535.53 100000

6. Conclusions. In table 1 it has been observed that acL.d of H and J 1trinagulations increases very fast as n increases and both remains nearly same

for 30n . However, their values remain parallel for 30n50. But values for J 1

become faster than H for 50n100. For J' triangulation, in beginning its valueincreases very slow rate but as n increases its value pick up very fast and becomevery close to a.d.d. of K or J1. Rate of increase of a.d.d. of A*K triangulation is the

slowest of all the triangulations. In the same way rate for K is faster than K or

J1 but less than H and J 1.From these observations one can conclude that those triangulations whose

a.d.d. increases at faster rate are considered to be inferior than the other. On thisbasis H can be considered inferior among all the triangulations and A*K is superioramong all the triangulations.

Again if we take a graphical representation of tabulated values of a.d.d. fordifferent triangulations, one can easily observed that for n15 there is not muchdeviation in the different graphs of a.d.d. for most of the triangulations. However,between n=15 and 20 the deviation between the value of H and K became faster,

but that of H and J remain the same, that is H and J 1 begins to deviate at the

faster rate than K . Deviation of A*K is not much from its previous values beforen=15. In addition to this one can also easily observed that K (or J1) and J' remain

106

Ch

art-

I fo

r co

mp

aris

ion

of

a.d

.d o

f th

e d

iffe

ren

ttr

ian

gula

tion

s fo

r va

riou

s va

lus

of n

107

Ch

art

II-c

omp

aris

ion

of

asym

pto

tic

a.d

.d. o

f th

e d

iffe

ren

ttr

ian

gula

tion

s fo

r va

riou

s va

lues

of

n

108

nearly same for values of n20 and their deviation is also less than H, J 1 and

K , but greater than A*K. However, from n=30 and onwards the values of a.d.d.

for triangulations H and J 1 as well as K steeply deviated from all othertriangulations. Whereas the deviation between A*K and that of K(or J1) and J' isnot much upto n=30 but deviation between them pick-up between 30 and 50 therebythe deviation of K (or J1) and J' became steeper than A*K. Though A*K is superiorof all the triangulations. But in practice K (for J1) is found to be much convenientto determine the fixed point of given mapping.

Table-2 compares the asymptotic a.d.d. for different triangulations andchart-II gives their graphical representation of asymptotic a.d.d. based on tabulatedvalue of table-2. Both tabulated values as well as graphical representation againhighlight, A*K as superior among all the triangulations.

REFERENCES[1] B.C. Eaves, Homotopy for computation of fixed points, Maths Programming 3 (1972),

1-22.[2] B.C. Eaves and R. Saigal, Homotopy for computation of fixed points on unbounded

regions. Maths Programming 2 (1972), 255-287.[3] H.W. Kuhn and J.G. Mackinnow, The sandwich method for finding fixed points.

J. Optimization Theory and Applications, 17 (1975).[4] O.H. Merril, Application andExistence of an Algorithm that Computes Fixed Point of

Certain Upper Semi Continuous Point into Set Mapping. University of Michigan, Ph.D.Dissertation (1972).

[5] R. Saigal, Investigations into the Efficiency of Fixed Point Algorithm Fixed Point;Algorithm and Applications, S.Karmardian (ed.) Academic Press.On the convergence of rate of algorithms for solving equations that are based on methodof complementary pivoting, Math. Operations Res., 2 (1977), 108-124.

[6] H.E. Scarf, The approximation of fixed points of a continuous mapping, SIAM J. Appl.Math. 155 (1967), 1328-1343.

[7] H.E. Scarf and T. Hansen, The computation of Economic Equilibria.Yale UniversityPress, New Haven and London (1973).

[8] M.J. Todd, On Triangulation for computing fixed points, Maths Programming, 10 (1976),322-346.The Computation Fixed Point and Application, Lecture Notes in Economics andMathematical System 124, Springer Verlag New York (1976).Improving the convergence of fixed point algorithms, Mathematical Programming Study,7 (1978), 151-169.Traversing large pieces of linearity in algorithms that solve equations by followingpiecewise-linear paths, Math. Oper. Res., 5 (1980), 242-257.Union Jack; Triangulations, Fixed Point; Algorithm and Applicatins, S. Karmardian(ed.) Academic Press (1974).

9. G. Lann Vander and A.J.J. Talman, An improvement of fixed point algorithms by usinga good triangulation, Math. Programming, 18 (1980) 274-285.

109


ON SOME GENERALIZED FRACTIONAL DERIVATIVE FORMULASBy

R.C. Singh Chandel and Vandana Gupta

Department of Mathematics, D.V. Postgraduate College, Orai-285001Uttar Pradesh, India

E-Mail:[email protected]

(Received : March 11, 2010)

ABSTRACTThe purpose of the present paper is to derive a number of key formulas for

fractional derivatives of generalized multiple hypergeometric functions of severalvariables, multivariable H-function and genralized multivariable polynomials.Each of these formulas can be shown to yield interesting new results for variousclasses of generalized hypergeometric functions of several variables. Some of theapplications of the new formulas provide potentially useful generalizations ofknown results in the theory of fractional calculus. Our results include all theresults of Chandel-Kumar [12] as special cases and all recent results of Ram-Chandk [32] as special cases for the Fox-Wright generalized hypergeometricfunction of Wright [54].2010 Mathematics Subject Classification : Primary 33C99; Secondary 26A33.Keywords: Generalized Fractional derivatives, generalized multiplehypergeometric functions, Multivariable H-function Generalized multivariablepolynomials, Fox-Wright generalized hypergeometric function, Bessel-Maitlandfunction, Wright generalized Bessel function, Mittag-Leffler function.

1. Introduction. The theory and applications of fractional calculus are

based largely upon the familiar differential operator xD defined by (cf., e.g., [31,

p.49]; see also [47, p.356])

(1.1)

1

0

1 0

0

x

x mm

xm

x t f t dt Re

D f xd

D f x Re m,m Ndx

where 0 0 1 2 3N N ,N , , ,... .

For 0 , equation (1.1) defines the classical Riemann Liouville fractional

derivative (or integral) of order or . On the other hand, when , the

110

equation (1.1) may be identified with the definition of the familiar Weyl fractional

derivative (or integral) of order or (see for details, [16, Chap. 13] and [34]).

For the sake of simplicity, the special case of the fractional calculus operator xD

when 0 is written as xD . Thus we have

(1.2) 0x xD D C .

For 0 1; , ,x R;m N , the generalized modified fractional derivative

operator due to Saigo is defined in Samko, kilbas and Marichev [34] as

(1.3)

0 0

1 1 11

mx, , m m m m

,x,m

d zD f x x t F , ; ; t / x

dz mf t dt .

The Multiplicity of m mx t in above equation is removed by requiring m mlog x t

as real for 0m mx t and is assumed to be well defined in the unit disk.

It is remarkable that

(1.4) 0 1, ,,x , xD f x D f x ,

where xD is the familiar Riemann-Liouville fractional derivative operator defined

by Miller and Ross [28]

For 0 1,m N ; , ,x R ; 0max , , the refined form due to

Bhatt and Raina [2] is given by

(1.5)

1 1

0m m, ,

,x ,mD x x .

Making an appeal to the Saigo modified fractional derivative operator 0, ,

,x ,mD ,

Miller and Rasso [28] investigated fractional derivative formulas. Kilbas [21]established the fractional integral formulas for the Wright ([54]; see also Erdélyi

[14]) introduced the generalized hypergeometric function p q Fox-Wright

generalized hypergeometric function, defined by

(1.6)

1 1

01

1

p

i i ki i ,p ip q p q q

kj j ,q j jj

a A ka ,A zz z

k!b ,B b B k

111

where

1 1

0 0 1 0q p

i j i j ij i

a ,b C; A ,B ; A ; 0 1 1i jA ,B i ,..., p; j ,..,q , for

suitable bounded values of z .

For details of conditions of its existence and the H-function due to Mathaiand Saxena [26], see Kilbas [21].

Wright [55] also introduced the special case of (1.6) (called Wright function)defined in the form:

(1.7)

0 10

1 ,

k

k

z, ; z z

, k k!

where ,z C and R .

Kiryakova [23] introduced a function J z called Bessel-Maitland function

or Wright generalized Bessel function defined by

(1.8)

0

11

1

k

k

zJ z v , ; z .

k k!

Other special case of (1.4) but generalizing the classical Mittag-Lefflerfunction (Erdélyi [16]) is given by Kilbas et al. [22].

Srivastava and Garg [38] introduced a general class of multivariablepolynomials defined by

(1.9) 1 1 1

1

11

11 1

0 1

s s s

s

s ss

k k ... h k L kkh ,...,h sL s shk ... h k

k ,...,k s

x xS x ,...,x L A L;k ,...,k ...

k ! k !

where 1 sh ,...,h are arbitrary positive integers and the coefficients 1 sA L;k ,...,k ,

0 1iL;k N ;i ,...,s are arbitrary constants real or complex, 0 0N N .

It is clear that for s=1, the polynomials (1.9) reduce to the polynomials ofSrivsatava [37] defined by

(1.10)

0

0

0 1 2l / h

h khkL l,k

k

lS x A x l N , , ,...

k!,

where h is arbitrary positive integer and the coefficients are arbitrary

constants, real or complex.The computation of fractional derivatives (and fractional integrals) of

112

special functions of one and more variables is important from the point of view ofthe usefulness of these results in (for example) the evaluation of series and integrals(cf., e.g., [29] and [52]), the derivative of generating functions [46, Chap. 5] andsolutions of differential and integral equation (cf. [29] and [39], Chap. 3. see also[27, 30 and 51]. Making an appeal to the operator (1.4), Chandel and Vishwakarma([6],[7]) have obtained fractional derivatives of confluent forms due to Chandel-

Vishwakarma [5]) of Karlsson's multiple hypergeometric function 20k nCDF and

other multiple hypergeometric functions of Lauricella [24], Chandel [3], Chandeland Gupta [4] including their confluent forms. Srivastava and Goyal [43] derivedfractional derivatives of the multivariable H-function of Srivastava and Panda([48]-[50]). Srivastava, Chandel and Vishwakarma [40] obtained several newfractional derivative formulas involving the multivariable H-function defined bySrivastava and Panda (see [50, p.271, eq. [4.1] et Seq.]) and studied systematicallyby them (see [48] [50] also [44]).

Further for special interest, Chandel and Vishwakarma [8] and Chandel-Sharma [11] derived fractional derivatives involving hypergeometric functions offour variables defined by Exton [18] and Sharma-Parihar [35], while Chandel-Sharma [11] established fractional derivative formulas for their own hypergeometricfunctions of four variables ([9],[10]). Recently, Chandel and Kumar [12] derivedgeneralizations and unifications of various key formulas of Srivatava Chandeland Vishwakarma [40]. Very recently employing the operator (1.3), Ram andChandak [32] derived a generalized derivative formula involving the product of

Fox-Wright generalized hypergeometric function p q defined by (1.6) and a general

class of multivariable polynomials defined by (1.9). Some special cases are alsodiscussed.

In the present paper, employing the operator (1.3), with the motivation ofChandel-Kumar [12] and Ram-Chandak [32], we derive generalizations andunifications of various key formulas of Chandel-Kumar [12] and Ram-Chandak[32]. Each of these formulas can be shown to yield interesting new results forvarious classes of generalized hypergeometric functions of several variables andgenearlized hypergeometric functions of one variable. Some of applications of thekey formulas provide potentially useful genearlizations of known results in thetheory of fractional calculus. Some special cases are also discussed.

2. Key Formulas. In this section, making an appeal to the result (1.5), wederive the following key formulas on generalized fractional derivatives involvingmultiple hypergeometric function of Srivastava-Daoust ([41],[42]; also seeSrivastava-Manocha [46, p.64 (18), (19), (20)]), multivariable H-function of

113

Srivsatava-Panda ([48]-[50]; see also Srivastava, Gupta-Goyal [44]), generalizedmultivariable polynomials due to Srivastava-Garg [38]:

(2.1) 11 11 1 1 1 1

1 1

1 10 0 1 1 1

rr rr r r r r

r r

m m, , , , m m,x ,m ,x ,m r r rD ...D x x ...x x

n

n

n n n

A:B';...;B

C:D';...;D n n n

a : ',..., : b' : ' ;...; b : ;F

c ; ',..., : d ' : ' ; ...; d : ;

11 1 1 1

1 1 1 1

'm ' mz x x 1

1 1 1 11 1 1

n nn nr r

r r r r r r r r'm ' m m m m m

r r r n r r r...x x ,...,z x x ...x x

1 3 0 0

3 0 01

0 0

0 0

nj j j j

n

nr

j j j j m A r: ';...; ; ;...;j j C r:D';...;D ; ;...; n

j j j j j j

a : ',..., , ,..., ,x F

c : ',..., , ,..., ,

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0

0 0 0 0 0 0

n n nr r r r

n n nr r r r

: ' ,..., , , ,..., , : ' ,..., , , ,..., ,..., : ' ,..., , ,..., , ,

: ' ,..., , , ,..., , : ' ,..., , , ,..., ,..., : ' ,..., , ,..., , r ,

1 1 1

1 1 1

0 0 1 0 0 0 0 1

0 0 0 0 0 0

n n nr r r r r r r r r

n n nr r r r r r r r r

: ' ,..., , ,..., , , : ' ,..., , , ,..., ,..., : ' ,..., , ,..., , :

: ' ,..., , ,..., , , : ' ,..., , ,..., ,..., : ' ,..., , ,..., :

11

11

r

n nm m

rn

n n' r

b' : ' ;...; b : ; ;...; ; x xZ ,...,Z , ,..., ,

d ' : ;...; d : ; ;...; ;

where

n

nA:B';...;

C:D';...;DF is generalized multiple hypergeometric function of Srivastava and

Daoust ([41],[42]), 0 1 0i i i i i i i i;m N ; , x R; max , ,

1 1

1

1

1

i ir r

i ir

m mi r

i

r

z x ...xZ ,

... 1i ,...,n;

11 1 1

1 0i iC D A B

i i i ij j j j

jj j j

, i=1,...,n

114

and

1 1 1 1

0i iC D A B

i i i ij j j j

j j j j

1i n ,...,n r .

(2.2) 11 11 1 1 1 1

1 1

1 10 0 1 1 1

rr rr r r r r

r r


0 n n

n n

n n n

, : , ;...; ,

A ,C: B ,D ;...; B ,D n n n

a , ',..., : b' : ' ; ...; b : ;H

c : ',..., : d ' : ' ;...; d : ;

1 11 1 1 1 1 1 1 1

1 1 1 1 1 1 1

n nn nr r

r r r r r r r r' 'm ' m m ' m m m m m

r r n n r r rz x x ...x x ,...,z x x ...x x

11 1

1 1 11

1

1 1111 1

0 1

rr r

i r rr

r

Nm mr rmm

r rN ,...,N r

x / x /... x ...x ...

N ! N !

1 1 1 1 10 3

3 31 1 1 1 1 1

1

1

n n

n n

n n, r: ', ' ;...; ,

A r ,C r: B',D' ;...; B ,D n n

a : ',..., , N : ' ,..., ,H

c : ',..., , N : ' ,..., ,

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1 1 1

1 1 1

n n nr r r r r

n n nr r r r r r

N : ' ,..., , N : ' ,..., ,..., N : ' ,..., ,

N : ' ,..., , : ' ,..., ,..., N : ' ,..., ,

1 1

1 1

n nn nr r r r r r r r r r r

n n n nr r r r r r r r r r

b' : ' ;...; b : ;N : ' ,..., , N : ' ,..., :

N : ' ,..., , : ' ,..., : d ' : ' ;...; d : ;

1 nZ ,...,Z

provided that 0 1 0i i i i i i i i,m N ; ,x R; Max , ,

1 11 1 1 1 1 0r rm m i i i i

r r r r rmax arg x ,...,arg x ,min ,..., ; ,..., ; ,..., ,

1 1

1

1

1

i ir r

i ir

m mi r

i

r

z x ...xZ

...

, 1i ,...,n and where

0 n n

n n

, : ', ' ;...; ,

A ,C: B',D' ;...; B ,DH

is multivariable H-

115

function due to Srivastava and Panda ([48]-[50]).

(2.3) 11 11 1 1 1 1

1 1

1 10 0 1 1 1

rr rr r r r r

r r

m m, , , , m x,x ,m ,x ,m r r rD ...D x x ...x x

1

1 1 1 11 1 1 1

n

n

n n n'm ' mA:B';...;B

C:D';...;D n nn

a : ',..., : b' : ' ;...; b : ;F z x x

c : ',..., : d ' : ' ;...; d : ;

11 1 1 1

1 1 1

n nnr r

r r r r r r'm m m m m mn

r r r r n r r r r...x ' x ,...,z x x ...x x

1 1

1 1

1

sr r

s

s s s

k' m k' mE:F ';...;FrG:H ';...;H s s s

e : ',..., : f ' : ' ;...; f : ;F w x ...x ,...,

g : ',..., : h' : ' ;...; h : ;

1 11

s sr rk m k m

s rw x ...x

1

1

i

i i i

rmi i i i i

ii i i i i i

x

3 0 0

3 0 0

0 0 0 0 0 0

0 0 0 0 0 0

n s

n s

n s

A E r: ';...; ;F ';...;F ; ;...;

C G r:D';...;D ;H ';...;H ; ;...; n s

a : ',..., , ,..., , e : ,..., , ',..., , ,..., ,F

c : ',..., , ,..., , g : ,..., , ',..., , ,..., ,

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0 0 0 0

0 0 0 0

n s n s

n s n s

; ' ,..., ,k ,...,k , , ,..., , : ' ,..., ,k ,...,k , , ,..., ,...,

: ' ,..., ,k ,...,k , , ,..., , : ' ,..., ,k ,...,k , , ,..., ,...,

0 0 0 0

0 0 0 0

n s n sr r r r r r r r r r r r r r

n s n sr r r r r r r r r r r r r r r

; ' ,..., ,k ,...,k , ,..., , , : ' ,..., ,k ,...,k , ,..., , ,

: ' ,..., ,k ,...,k , ,..., , , : ' ,..., ,k ,...,k , ,..., , ,

1 1 1

1 1 1

0 0 1 0 0 0 0 1

0 0 0 0

n nr r r

n nr r r

: ' ,..., , ,..., , , ,..., ; ...; : ' ,..., , ,..., , :

: ' ,..., , ,..., ,..., : ' ,..., , ,..., :

1

n n s s

nn n s s

b' : ' ;...; b : ; f ' : ' ;...; f : ; ;...; ;Z ,...,Z ,

d ' : ' ;...; d : ; h' : ' ;...; h : ; ;...; ;

116

1 1 1 1 1 11 1 1 1 1

s sr r r r r rk' m k' m k m k m m m

r r r r rw x ...x ,...,w x ...x , x / ,..., x / ,

valid if 1 1

1

1

1

i ir r

i ir

m mi r

i

r

z x ...xZ ,

...

0 1 0i i i i i i i i,m N ; , ,x R; Max , ,

1 1 1 0 1i i i ir r rmin ,..., ; ,..., ; ;..., ,i ,...,n.

(2.4) 11 11 1 1 1 1

1 1

1 10 0 1 1 1

rr rr r r r r

r r


1

1 1 1 11 1 1 1

n

n

n n n'm ' mA:B';...;B

C:D';...;D n nn

a : ',..., : b' : ' ; ...; b : ;F z x x

c : ',..., : d ' : ' ;...; d : ;

11 1 1 1

1 1 1

n nn nr r

r r r r r r r r'm ' m m m m m

r r r n r r r...x x ,...,z x x ...x x

1 1 1 1 11 1 1

s ss r r r rh ,...,h k' m k' m k m k m

L r s rS w x ...x ,...,w x ...x

1

1 1 11 11 1 1 1 11

1 1 1 1 1

r

r r rm mr r r r rr

r r r r r

... ...x ...x

...

1 1

1

1 1 10

s s

s

h R ... h R L

s s sR ,...,R

L,h R ... h R A L : R ,...,R

1 1 1 1 1

1 1 1 1 1 1

s ss r r r s

s ss r r r r s

,k' R ... k R ... ,k' R ... k R

,k' R ... k R ... ,k' R ... k R

1 1 1 1 1 1 1

1 1 1 1 1 1 1

s ss r r r r r s

s ss r r r r r s

,k' R ... k R ... ,k' R ... k R

,k' R ... k R ... ,k' R ... k r

11 1 1 1

1 1 1

1

ss sr r r r

R Rk' m k' m k m k m

r s r

s

w x ...x w x ...x...

R ! R !

3 0 0

3 0 0

0 0

0 0

n

n

n

A r:B';...;B ; ;...;

C r:D';...;D ; ;...; n

a : ',..., , ,..., ,F

c : ',..., , ,..., ,

1 1 1 1

1 1 1

0 0 0 0

0 0 0 0

n nr r r r

n nr r r

: ' ,..., , , ,..., ,..., : ' ,..., , ,..., , ,

: ' ,..., , ,..., ,..., : ' ,..., , ,..., ,

117

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1

0 0 0 0

0 0 0 0

s n s ns r r r s r r r

s n s ns r r r r s r r r

k' R ... k R : ' ,..., , , ,..., ,..., k' R ... k R : ' ,..., , ,..., , ,

k' R ... k R : ' ,..., , , ,..., ,..., k' R ... k R : ' ,..., , ,..., , ,

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

0 0 0 0

0 0

s n s ns r r r r r s r r r

s n s ns r r r r r s r r

k' R ... k R : ' ,..., , , ,..., ,..., k' R ... k R : ' ,..., , ,..., , :

k' R ... k R : ' ,..., , , ,..., ,..., k' R ... k R : ' ,...,

0 0 r, ,..., , :

1 11

11

r r

n nm m

rn

n nr

b' : ' ; ...; b : ; ;...; ; x xZ ,...,Z , ,...,

d ' : ' ;...; d : ; ;...; ;

,

provided that 1 1

1

1

1

i ir r

i ir

m mi r

i

r

z x ...xZ

...

, 0 1 0i i i i i i i i,m N ; , ,x R; max , ,

1 1 1 1

1 0 1i iC D A B

i i i ij j j j

j j j j

,i ,...,n

;

1 1 1 1

0i iC D A B

i i i ij j j j

j j j j

,

1i n ,...,n r .

n

nA:B';...;B

C:D';...;DF is generalized multiple hypergeometric function of Srivastava and Daoust

([41],[42]] while 11

sh ,....,hL sS x ,...,x are generalized multivariable polynomials due to

Srivastava and Garg [38], defined by (1.9).

(2.5) 11 11 1 1 1 1

1 1

1 10 0 1 1 1

rr rr r r r r

r r


1 10 n n

n n

n n n

, : , ;...; ,

A ,C: B',D' ;...; B ,D n n n

a : ',..., : b' : ' ; ...; b : ;H

c : ',..., : d' : ' ;...; d : ;

11 11 1 1 1 1 1 1 1

1 1 1 1 1 1 1

n nn nr r

r r r r r r r r' 'm ' m m m m m m m

r r r n r r rz x x ...x x ,...,z x x ...x x

1 1 1 1 11 1 1

s ss r r r rh ,...,h k' m k' m k m k m

L r s rS w x ...x ,...,w x ...x

118

1 1 11 1 1 1 1 1

1

11 1 1 1

0 1

s s s ss

s

h R ... h R L RRN k R ... k R ms

s s sR ,...,R s

w wL,h R ... h R A L;R ,...,R ... x ...

R ! R !

1 0 3

3 3

n nsr r r r s r

n n

n, r: ', ' ;...; ,N k' R ... k R m

r A r ,C r: B',D' ;...; B ,D n

a : ',..., ,x H

c : ',..., ,

1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

1 1

1 1

n s ns

n s ns

N : ' ,..., , N k' R ... k R : ' ,..., ,

: ' ,..., , N k' R ... k R : ' ,..., ,

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

1

1

s ns

s ns

N k' R ... k R : ' ,..., ,...,

N k' R ... k R : ' ,..., ,...,

1

1

1 1

1 1

n s nr r r r r r r r r s r r

n s nr r r r r r r r r s r r

N : ' ,..., , N k' R ... k R : ' ,..., ,

: ' ,..., , N k' R ... k R : ' ,..., ,

1

1

1

1

s nr r r r r r r s r r

s nr r r r r r r s r r

N k' R ... k R : ' ,..., :

N k' R ... k R : ' ,..., :

1

n n

nn n

b' : ' ;...; b : ;Z ,...,Z

d' : ' ;...; d : ;

,

where 1 1

1

1

1

i ir r

i ir

m mi r

i

r

z x ...xZ ,

...

1 1 1 0i i i i

r r rmin ,..., ; ,..., ; ,..., , 0 1j j,m N ;

0 1j j j j j j, ,x R; max , , j ,...,r and 11

sh ,...,hL sS w ,...,w are generalized

multivariable polynomials due to Srivastava and Garg [38] defined by (1.9). Ourresult (2.5) among others also includes (2.4) as special case.

3. Proofs of the Key Formulas. In this section, we prove the results ofSection 2.Proof of (2.1). For brevity, we denote

1

1 11 1 1

01 1

1 1 1

1

n

n

n

A B' Bn n n

j j n j j j j n jj j jC D' DnM ,...,M n n n

j j n j j j j n jj j j

a ,M ' ... M b' ,M ' ... b ,MS .

M !...M ! c ,M ' ... M d' ,M ' ... d ,M

119

Therefore, left hand side of (2.1)

1 1 1 11 1 1 1 11 1 1 1 1 1

1 1

1

1 0 0 1 1 1

nnnnn r r r

r r

' M ... M' M ... M mMM , , , , mn ,x ,m ,x ,mSz ...z D ...D x x ...

11 1nn

r r r nr r r n r r r' M ... M' M ... M m m

r r rx x

11 1 2

1 1 1 11

11 1

0 11

1 1 rn

n nn r r n

r

N NMMrn r

' M ... M ' M ... MN ,...,N rr

/ /z ...z ...S ...

N ! N !...

1 1 1 1 1 1n n

n r r n r r' M ... M ,N ... ' M ... M ,N

1 1 1 1 1 1 1 11 1 1

1 1

1 1

0 0 1

n nn r r r r r r rr r r

r r

' M ... M N m ' M ... M N m, , , ,,x ,m ,x ,m rD ...D x ...x

1

1 1

1 10 1

j

j j j

r

nr rmj j j j j j j j n j j

j nj jN ,...,Nj j j j j j j j j n j j

, ' M ... M Nx S

, ' M ... M N

11 1

1 1

1n nj j j j j n j j j j j n

n nj j j j j n j j j j j n

, ' M ... M N , M ... M

, ' M ... M N , ' M ... M

1 11 1 1 1 1

1 1

1 1 1 1

1 1 1

1 1

n n n rr r r r r

n nr r

M M N N' m ' m m m m mr n r r

' 'r r r

n r

z x ...x z x ...x x x... ...

... ...M ! M ! N ! N !

[By making an appeal to (1.5)],which can be written in the form of right hand side of (2.1).Proof of (2.2). For brevity, we denote

1

1 1

1 11 1

11

2 1n

ni

j j ij i

n r rA CL Li i

j j i j j ij ji i

aI ...

w a c

1 1

1

11

1

1

i i

i i

i

i i i ij j i j j in

j j

D Bi i i i i

j j i j j ijj

d b.

d b

120

Therefore, left hand side of (2.2)

1 1

1 1 1 1

1 1

1

n r

n nn r r n

n r' ... ' ...

r

z ...z ...I

...

1

1

1

1 1 1 1 1 1

0 1 1

1 r

r

r

N ... N n nn r r n r r

N NN ,...,N r r

' ... ,N ... ' ... ,N

... N ! ...N !

1 1 1 1 1 1 1 11 1 1

1 1

1 1

0 0 1

n nn r r r n n r rr r r

r r

' ... N m ' ... N m, , , ,,x ,m ,x ,m rD ...D x ...x

11 1

1 1 1 11

11 1

0 11

1 1 rn r

n nn r r n

r

N N

rn r' ... ' ...

N ,...,N rr

z ...z ...I ...

N ! N !...

1 1 1 1 1 1

1 1 1 1 1

n nn r r r r n

n nn r r r n

N ' ... N ' ......

' ... ' ...

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

n nn n

n nn n

' ... ' ... N

' ... N ... ' ... N

...

1 1

1 1

n nr r r n r r r r r n r r r r

n nr r r n r r r r r r n r r r r

' ... ' ... N

' ... N ... ' ... N

1 1 1 1 1 1 1 1 11 1

1

n nn r r r r n r r r r r' ... N m ' ... N m

rx ...x (By an use of (1.5)],

which can be reformed in the form of right hand side of (2.2).Proof. of (2.3). For brevity, if we denote

11 1

1 1 1 11

11 1

0 11

1 1 rn r

n nn r r n

r

N NMMrn r

' M ... M ' M ... MN ,...,N rr

/ /z ...z ...K S ...

N ! N !...

1 1 1 1 1 1n n

n r r n r r' M ... M ,N ... ' M ... M ,N

1

1

1 11 1 1 1

0 11 1

1 1 1

s

s

s

s

E F ' Fs s s

j j s j j j j s j RRj j j sG H ' HsR ,...,R ss s

j j s j j j j s jj j j

e ,R ' ... R f ' ,R ' ... f ,Rw w

... .R ! R !g ,R ' ... R h' ,R ' ... h ,R

121

The left hand side of (2.3)

1 1 1 1 1 1 1 1 1 1 1 11 1 1

1 1

1 1

0 0 1

n s n sn s r r r n r r r r s rr r r

r r

' M ... M N k' R ... k R m ... ' M ... M N k' R ... k R m, , , ,,x ,m ,x ,m rKD ...D x x

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

n sn s

n sn s

, ' M ... M N k' R ... k RK

, ' M ... M N k' R ... k R

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

n sn s

n sn s

, ' M ... M N ... k' R ... k R...

, ' M ... M N ... k' R ... k R

1 1

1 1

n sr r r n r r r r s

n sr r r r n r r r r s

, ' M ... M N k' R ... k R

, ' M ... M N k' R ... k R

1 1

1 1

n sr r r r r n r r r r s


, ' M ... M N k' R ... k R

, ' M ... M N k' R ... k R

1 1 1 1 1 1 1 1 1 1 1 1 11 1

1

n s n sn s r r r r n r r r r s r' M ... M N k' R ... k R m ' M ... M N k' R ... k R m

rx ...x

[By making an appeal to (1.5)],which can be adjusted in the form of right hand side of (2.3).Proof of (2.4). For brevity considering

1 1

1 1 1 11

1 11 1 1 1 1

01

n r

n nn r r n

r

MMnn r

n' M ... M ' M ... MN ,...,Nr

z ...z ...M S ' M ... M ,N ...

...

1

11

1

1 1 rN Nrn

r r n r rr

/ /' M ... M ,N ...

N ! N !

1 1 1

1 11

11

0 1

s s s

s ss

h R ... h R L RRs

sh R ... h RR ,...,R s

w wL A L; R ,...,R ... .

R ! R !

Left hand side of (2.4)

1 1 1 1 1 1 1 1 1 1 1 11 1 1

1 1

1 1

0 0 1

n s n sn s r r r n r r r r s rr r r

r r

' M ... M N k' R ... k R m ' M ... M N k' R ... k R m, , , ,,x ,m ,x ,m rMD ...D x ...x

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

n sn s

n sn s

, ' M ... M N k' R ... k RM

, ' M ... M N k' R ... k R

122

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

n sn s

n sn s

, ' M ... M N ... k' R ... k R...

, ' M ... M N ... k' R ... k R

1 1

1 1



, ' M ... M N k' R ... k R

, ' M ... M N k' R ... k R

1 1

1 1



, ' M ... M N k' R ... k R

, ' M ... M N k' R ... k R

1 1 1 1 1 1 1 1 1 1 1 1 11 1

1

n s n sn s r r r r n r r r r s r' M ... M N k' R ... k R m ' M ... M N k' R ... k R m

rx ...x

[By making an appeal to (1.5)],which can be reformed as right hand side of (2.4).Proof of (2.5). We can writeLeft hand side of (2.5)

11 1

1 1 1 11

11 1

0 11

1 1 rn r

n nn r r n

r

N N

rn r' ... ' ...

N ,...,N rr

/ /z ...z ...I ...

N ! N !...

1 1 1 1 1 1n n

n r r r n r' ... ,N ... ' ... ,N

1 1 1

1

11 1 1

0 1

s s s

s

h R ... h R L RRs

s s sR ,...,R s

w wL,h R ... h R A L; R ,...,R ...

R ! R !

1 1 1 1 1 1 1 1 1 1 1 11 1

1

n s n sn s r r r n r r r r s r' ... N k' R ... k R m ' ... N k' R ... k R m

rx ...x

1 11 1

1 1 1 11 1 0

1 11 1 1

0 01

s sn r

n nn r r n

r

h R ... h R Ln r

s s s' ... ' ...N ,...,N R ,...,Rr

z ...z ...I L,h R ... h R A L;R ,...,R

...

1111

1 1

1 1 rsN NRR

rs

s r

/ /ww... ...

R ! R ! N ! N !

1 1 1 1 1 1

1 1 1 1 1

n nn r r r n r

n nn r r r n

' ... N ... ' ... N

' ... ... ' ...

1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

n sn s

n sn s

' ... N k' R ... k R

' ... N k' R ... k R

123

1 1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 1

n sn s

n sn s

' ... N k' R ... k R...

' ... N k' R ... k R

1 1

1 1



' ... N k' R ... k R

' ... N k' R ... k R

1 1

1 1



' ... N k' R ... k R

' ... N k' R ... k R

1 1 1 1 1 1 1 1 1 11 1

1

n s n sn r r r s r r r r n r r r r s r' ... N k' R ... k R m ' ... N k' R ... k R m

rx ...x

[By making an application of (1.5)],which can be expressed in the form of right hand side of (2.5).

3. Special Cases. In this section, we mention the special cases of ourresults.Special Cases of (2.1)

(a) For 1 1i i im , ,i ,...,r ; (2.1) reduces to Chandel and Kumar [12,p.107, (2.1)].

(b) For 1 1 11 1r ,m , , (2.1) gives Srivastava, Chandel and Vishwakarma

[40, p. 567 (3.1)].

(c) For 1 2 1 1 2 22 1r ,m m , , , (2.1) reduces to Srivastava, Chandel and

Vishwakarma [40, p. 568 (3.3)].Special Cases of (2.2)

(a) For 1 1i i im , ;i ,...,r ; (2.2) reduces to Chandel and Kumar [12, p.108

(2.3)], which also generalizes the earlier results due to Srivastava, Chandeland Vishwakarma [40, p. 563 (2.1), p. 564 (2.3)].

Special Cases of (2.3).

(a) For 1 1i i im , ,i ,...,r ; (2.3) reduces to the result due to Chandel and

Kumar [12, p.107, (2.2)].

(b) For 1 1 11 1r ,m , , (2.3) reduces to the improved version of the result due

to Srivastava, Chandel and Vishwakrama [40, p. 567 (3.2)](c) Due to general nature of the functions involved in (2.3), specializing the number

of variables and other parameters in (2.3), we can get several interesting knownand unknown results.

Special Cases of (2.4).

(a) For 1 11 0 1 0r , ,n , ' , the result (2.4) will include the result due to Ram

124

and Chandak [32, p.53 (10)] involving Fox-Wright generalized hypergeometric

function p q and generalized multi-variable polynomials 1 sh ,...,hLS of Srivastava

and Garg [38].

(b) For 1 11 0 1 0r , ,n , ' and 1 1 , (2.4) includes an interesting result

due to Ram-Chandak [32, p. 53 (11)] for the Riemann-Liouville derivativeoperator-defined by Miller and Ross [28].

(c) For s=1, the polynomials 11

sh ,...,hL sS x ,...,x reduce to the Srivastava polynomials

hlS defined by (1.10), therefore, for 1 1 1 11 0 1 0r , ,n , ' , and 1 1s ,

(2.4) gives an interesing result due to Ram-Chandak [32, p. 54(12)] involvingSrivastava polynomials.

(d) Similarly all the following results due to Ram and Chandak [32, p.54 (13),

(14), (15)] are included in our result (2.4). Since for 0 00 1 1k, ll ,A ,S x ,

therefore our result (2.4), also gives the result due to Kilbas [21] as specialcase of Ram and Chandak [32 (15)]

(e) Since Wright function , ; z defined by (1.7) and Wright generalized Bessel

function J z defined by (1.8) are special cases of Fox-Wright generalized

hypergeometric function p q z defined by (1.6), therefore, all the results due

to Ram and Chandak [32, pp. 55-57 (16), (17), (18),(19)(20),(21)] are included inour result (2.4) as special cases.

Special Cases of (2.5). Our result (2.5) includes (2.4) along with all results ofRam and Chandak [32] as special cases.

4. Other Special Cases as Application of one Fractional Derivative.In this section, making an appeal to (1.5) and one fractional derivative we derive

(4.1)

11

0 1

nn

n

n n n

m mm, , A:B';...;B,x ,m nC:D';...;D n n n

a : ',..., : b' : ' ;...; b :D x S y x ,..., y x

c : ',..., : d' : ' ;...; d :

1 11 2

21 1

n

n

nn nm A : ';...;

C :D';...;D nn n

a : ',..., , : ,..., , : ,..., :x S

c : ',..., , : ,..., , : ,..., :

125

1

1n

n n

mmn

n n

b' : ' ;...; b :y x ,..., y x

d' : ' ;...; d :

,

provided that 0 1 0 1i,m N ; , , ,x R, max , and

1 1 1 1

1 0i iC D A B

i i i ij j j j

j j j j

i=1,...,n.

(4.2) 10 1 1

m n, , m m,x,m D n nD x F ,b ,...,b ; ; y x ,..., y x

1

1 1

mn m m

D n n

xF ,b ,...,b ; ; y x ,..., y x

valid if 0 1 0 1,m N ; , ,x R,k max , , 1 1 1ny x ,..., y x ,

where nDF is well known Lauricella function [24].

(4.3) 10 1 1

m n, , m m,x,m D n nD x F ,b ,...,b , ; y x ,..., y x

1

1 1

mn m m

D n n

xF ,b ,...,b ; ; y x ,..., y x

,

provided that all conditions of (4.2) are satisfied.

(4.4) 10 1 1

m n, , m m,x,m D n nD x F ,b ,...,b , ; y x ,..., y x

1

1 1

mn m m

D n n

xF ,b ,...,b ; ; y x ,..., y x ,

where all conditions of (4.2) hold true.

(4.5) 10 1 1

m n, , m m,x,m D n nD x F ,b ,...,b ; ; y x ,..., y x

1

1 1

mn m m

D n n

xF ,b ,...,b ; ; y x ,..., y x

,

valid if all conditions of (4.2) are satisfied.

(4.6) 10

m, , m,x ,mD x exp x

126

1

2 2

mmx

F , ; , ; x ,

provided that 0 1 1,m N, , ,x R, max , .

(4.7) 10 2 2

m, , m,x ,mD x F , ; , ; x

1 mmx

exp x ,

where all conditions of (4.6) are satisfied.

(4.8) 10 1 1

m, , m,x ,mD x F ; ; x

1

1 1

mmx

F ; ; x

,

which holds true if all conditions of (4.6) are satisfied.

(4.9) 10 1 1

m, , m,x ,mD x F ; ; x

1

1 1

mmx

F ; ; x

,

where all conditions of (4.6) hold true.

(4.10) 10 1 1

m, , m,x ,mD x F ; ; x

1

1 1

mmx

F ; ; x

,

provided that all conditions of (4.6) hold time.

(4.11) 10 1 1

m, , m,x ,mD x F ; ; x

1

1 1

mmx

F ; ; x

,

which is true if all conditions of (4.6) are setisfied.

(4.12) 10 2 1

m, , m,x ,mD x F ; ; x

127

1

1m

mxx

valid if all conditions of (4.6) are satisfied.

(4.13) 10 2 1

m, , m,x ,mD x F ; , ; x

1

1m

mxx

,

where all conditions of (4.6) are satisfied.

(4.14) 10 2 1

m, , m,x ,mD x F ; , ; x

1

1m

mxx

,

provided that all conditions of (4.6) hold true.

(4.15) 10 1 2

m, , m,x ,mD x F ; , ; x

1

1m

mxx

,

valid if all conditions of (4.6) are true.

(4.16) 10 1 2

m, , m,x ,mD x F ; , ; x

1

0 1

mmx

F ; ; x

,

which holds true if all conditions of (4.6) are satisfied.REFERENCES

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variables, J. Math. Anal. Appl., 112, (1985), 641-651.[44] H.M. Srivastava, K.C. Gupta and S.P. Goyal, The H-functions of One and Two Variables

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Their Applications, Halsted Press Chichester and Wiley, New York/Chichester/Brisbane/Toronto, 1989.

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131


STRONG CONVERGENCE THEOREMS FOR UNIFORMLY EQUI-CONTINUOUS AND ASYMPTOTICALLY QUASI-NONEXPANSIVE

MAPPINGSBy

Gurucharan Singh Saluja

Department of Mathematics and Information TechnologyGovernment Nagarjuna Postgraduae College of Science, Raipur, Chhatishgarh,

IndiaE-Mail : [email protected], [email protected]

(Received : March 10, 2011; Revised : February 5, 2012)

ABSTRACTThe purpose of this paper is prove some strong convergence theorems of

the modified Ishikawa iterative sequences with errors for unformly equi-continuousand asymptotically quasi-nonexpansive mapping in the setup of unifromly convexBanach spaces by using condition (A) instead of comletely continuous ordemicompact condition. Our results improve and generalize the correspondingresults of Rhoades [6], Schu [7,8]. Tan and Xu [10,11] Xu and Noor [12] and manyothers.2000 Mathematics Subject Classification : 47H09, 47H10, 47J25.Keywords: Asymptotically nonexpansive and Asymptotically quasi nonexpansivemappings, Fixed points, The modified Ishikawa iterative sequence with errors,Strong convergence, Uniformly convex Banach space, Uniformly equi-continuousmapping, Uniformly Holder continuous mapping.

1. Introduction and Preliminaries. Let E be a real normed linear space,K be a nonempty subset of E. Throughout the paper, N denotes the set of positiveintegers and F(T)={x:Tx=x} the set of fixed points of a mapping T. Let T:KK bea given mapping.(1) T is said to be asymptotically nonexpansive [2] if there exists a sequence

[1, )nk with

lim 1nnk such that

,n nnT x T y k x y (1.1)

for all ,x y K and .n N

(2) T is said to be asymptotically quasi-nonexpansive if F T and there

exists a sequence {kn} in [1,) with

lim 1nnk such that

132

,nnT x p k x p (1.2)

for all ,x K p F T and .n N

(3) T is said to be uniformly L-Lipschitzian if there exists a positive constantL such that

,n nT x T y L x y (1.3)


(4) T is said to be uniformly Holder continuous [5] if there exist positiveconstants L and such that

n nT x T y L x y , (1.4)


(5) T is said to be uniformly equi-continuous [5] if, for any >0, there exists

0 such that

n nT x T y (1.5)

whenever x y for all ,x y K and 1n or, equivalently, T is uniformly equi-

continuous if and only if 0n nn nT x T y whenever 0n nx y as n .

Remark 1.1 (i) It is easy to see that, if T is asymptotically nonexpansive, then itis uniformly L-Lipschitzian.(ii) If T is uniformly L-Lipschitzian, then it is uniformly Holder continuous withconstants L > 0 and = 1.(iii) If T is uniformly Holder continuous, then it is uniformly equi-continuous.

This class of asymptotically nonexpansive mappings was introduced byGoebel and Kirk [2] in 1972. They proved that, if K is a nonempty boundedasymptotically nonexpansive self-mapping of K has a fixed point. Moreover theset F (T) of fixed points of T is closed and convex. Since 1972, many authors havestudied weak and strong convergence problem of the Mann and Ishikawa iterativesequences (with errors) for asymptotically nonexpansive mappings in Hilbertspaces and Banach spaces (see [2,6,7,8,10,11,12] and references therein).

Recenyly, Liu [3] studied modified Ishikawa iterative sequences with errorsfor asymptotically quasi-nonexpansive and unifornly Holder continuous mappingsin uniformly convex Banach spaces and established some strong convergencetheorems which extended some corresponding results of Tan and Xu [11].

The purpose of this paper is to extend and improve some resuits of [3] for

133

uniformly equi-continuous and asymptotically quasi-nonexpansive mappings. Alsoour results improve and generalize the corresponding results of [6,7,8,11,12,] andmany others.

In order to prove the main resuIts in this paper, we need the followinglemmas :

Lemma 1.1. (Tan and Xu [10]). 1 1,n nn n

Let

and 1n n

r

be sequences of

nonnegtive numbers satisfying the inequality

1 1 , 1.n n n n nr

If 1

nn

and 1

nn

r

, then nnLim

exists. In particular, 1n n

has a

subsequence which converges to zero, then 0nnLim

.

Lemma 1.2. ([1]) Let X be a uniformly convex Banach space and Br(0) be a closedball of X, Then there exists a continuous increasing convex function

: 0, 0,g with g(0)=0 such that

2 2 2 2x y z x y z g x y

for all , 0rx y B and , , 0,1 with 1 .

2. Main Results. Now, we give the main results of this paper.Lemma 2.1. Let E be a normed linear space and K be a nonempty convex subset ofE. Let T:KK be an asymptotically quasi-nonexpansive mapping with F(T).

and a seqeunce 1,nk with 1nnLimk

and 2

11nn

k

. Let {xn} be a

sequence in K defined by

,nn n n n n n ny a x b T x c v

1 , 1,nn n n n n n nx a x b T y c u n (2.1)

where {un}, {vn} are bounded sequences in E and {an},{bn},{cn},{ na }, { nb },{ nc }

are sequences in [0,1] and an+bn+cn= n n na b c =1 with the restrictions

1 nnc

and

1 n nnb c

. Then we have the following:

(i) nnLim x p

exists for any p F T ,

(ii) ,nnLimd x F T

exists, where d(x,F(T)) denotes the distance from x to the

set F(T).

134

Proof of (i). Let p F T . Since {un} and {vn} are bounded sequences in K. So we

can set

1 1

max sup ,sup .n nn n

M u p v p

Then it follows from (2.1) that

ny p nn n n n n na x b T x c v p

nn n n n n na x p b T x p c v p

nn n n n n na x p b T x p c v p

n n n n n n na x p b k x p c v p

n n n n n nna b k x p c v p

1 n n n n nc k x p c v p

.n n nk x p c M (2.2)

Again from (2.1) and (2.2), we have

1nx p nn n n n n na x b T y c u p

nn n n n n na x p b T y p c u p

nn n n n n na x p b T y p c u p

n n n n n n na x p b k y p c u p

n n n n n n n n na x p b k k x p c M c u p

2n n n n n n n na b k x p b k c M c M

21 n n n n n n nc k x p b k c M c M

2n n n n n nk x p b k c M c M

21 1n n n n n nk x p b c k c M (2.3)

since 2

1 11 ,n nn n

k c

and

1 n nnb c

, it follows from Lemma 1.1, we

know that lim nnx p

exists. This completes the proof of part (i).

Proof of (ii) From conclusion of part (i), we have

135

21 1 1 .n n n n n n nx p k x p b c k c M

This gives that

21, 1 1 , .n n n n n n nd x F T k d x F T b c k c M

Since 2

1 11 ,n nn n

k c

and

1 n nnb c

, it follows from Lemma 1.1,

we know that lim ,nnd x F T

exists. This completes the proof of part (ii).

Theorem 2.1. Let E be a uniformly convex Banach space, K be a nonempty convexsubset of E and T:KK be a uniformly equi-continuous and asymptotically quasi-

nonexpansive mapping with F(T) and a sequence 1,nk with lim 1nnk

and 2

11nn

k

. Let {xn}be a sequence in K defined by (2.1) with the following

restrictions:

(i) 0 1nb b and 1n nb b for all 1n .

(ii)1 nnb

, (iii) lim 0,nn

b

(iv) 1 nnc

and

1 n nnb c

.

Then liminf 0n nnx Tx

.

Proof. Since T:KK is asymptotically quasi-nonexpansive, we have

2nn n n n nT y p k y p k x p

for any p F T . By Lemma 2.1(i), we know that lim nnx p

exists. Hence nx p

and nnT y p are bounded sequences in E. Set 1 sup : 1 ,nr x p n

2 sup : 1 ,nnr T y p n 3 sup : 1 ,nr u p n 4 sup : 1nr v p n and

max : 1,2,3,4ir r i for any fixed p F T . Then we have , ,nn nx p T y p

, 0n n ru p v p B for all 1n . By using Lemma 1.2 and (2.1), we have

2ny p

2nn n n n n na x p b T x p c v p

22 2n nn n n n n n n n n na x p b T x p c v p a b g x T x

2 22 2n n n n n na x p b k x p c r

136

22 2n n n n na b k x p c r

22 21 n n n nc k x p c r

22 2.n n nk x p c r (2.4)

Again, using Lemma 1.2, (2.1) and (2.4) we obtain

21nx p

2nn n n n n na x p b T x p c u p

22 2n nn n n n n n n n n na x p b T y p c u p a b g x T y

2 2 22 nn n n n n n n n n n na x p b k y p c u p a b g x T y

2 22 2 2 2 nn n n n n n n n n n n na x p b k k x p c r c r a b g x T y

24 2 2 2 nn n n n n n n n n n n na b k x p b c k r c r a b g x T y

24 2 2 21 nn n n n n n n n n n nc k x p b c k r c r a b g x T y

24 2 2 nn n n n n n n n n nk x p b c k c r a b g x T y

24 2 21 1 nn n n n n n n n n nk x p b c k c r a b g x T y . (2.5)

Note that 2

11nn

k

is equivalent 4

11nn

k

and so, setting

2 4 1 ,n nr k then 1 nn

. Furthermore, since : 0, 0,g is a

continuous increasing convex function and nn nx T y is a bounded sequence in

E, we assert that nn ng x T y is bounded. Set n

n n n nc g x T y , we have

1 nn

. Since {kn} is bounded, and by hypothesis

1 n nnb c

so

2

1 n n nnb c k

. Now, set

2 2n n n n n n nb c k c r .

Then 1 nn

. By the assumption (i), we have (1–bn)(1–b). It follows from

(2.5) that2

1nx p 2 21 nn n n n n n n n n n nx p b c b g x T y b c k c r

137

2 2 21 nn n n n n n n n n n nx p b b g x T y b c k c r

21 n

n n n n n nx p b b g x T y (2.6)

which leads to

2 211 ,n

n n n n n nb b g x T y x p x p (2.7)

and

2 211 1 1 1 2 11 ,n

n n n n n nb b g x T y x p x p for all n1. (2.8)

Adding on both sides of (2.7) and (2.8) and using the condition 1n nb b for all

1n , we have

11 1 1

1

1 .n nn n n n n

n

b b g x T y g x T y

(2.9)

Since 1 nnb

by the assumption (ii), we have

11 1liminf 0n n

n n n nng x T y g x T y

. (2.10)

By virtue of the continity and monotonicity of function g, we assert that there

exists a subsequence {jnx } of {xn} such that

1

1 10, 0j j

j j j j

n nn n n nx T y x T y

as j . (2.11)

By the assumption (iii), we see that

0nn n n n n n n ny x b x T x c v x , as n . (2.12)

It follows from the uniform equi-continuity of T that

0n nn nT y T x as n . (2.13)

Now we observe thatn n n n

n n n n n nx T x x T y T y T x . (2.14)

It follows from (2.11) and (2.13) that

0j

j j

nn nx T x and 1

1 10j

j j

nn nx T x

, as j .

Since 0j

j j

nn nT x x as j , we have

10j

j j j j j j j j

nn n n n n n n nx x b T y x c u x

as j . (2.15)

138


10j

j j

nn nT x x

as j . (2.16)

Again, from above inequalities, we observe that

1 1

j

j j

nn nT x x

1 1

j j j

j j j j

n n nn n n nT x T x T x x

1 1

0j j j

j j j j j j

n n nn n n n n nT x T x T x x x x

as j . (2.17)


1 1

1 0j

j j

nn nT x Tx

as j , (2.18)

and

1 1

1 1 1 1 1 1

j j

j j j j j j

n nn n n n n nx Tx x T x T x Tx

. (2.19)

Therefore, it follows from (2.17), (2.18) and the above inequality that

1 10

j jn nx Tx as .j (2.20)

This completes the proof.Let {zn} be a given sequence in K. Recall that a mapping T:KK with

F T is said to satisfy condition (A) [9] if there exists a nondecreasing function

with : 0, 0,f with f(0)=0, f(r)>0 for all 0,r such that

,n n nz Tz f d z F T for all 1n ,

where , inf :n nd z F T z p p F T .

By using Theorem 2.1, we have the following :Theorem 2.2. Let E be a uniformly convex Banach space, K be a nonempty convexsubset of E and T:KK be a unifromly equi-continuous and asymptotically quasi-

nonexpansive mapping with F T and a sequence 1,nk with lim 1nnk

and 2

11nn

k

. Let {xn} be a sequence in K defined by (2.1) with the following

restrictions:

(i) 0 1nb b and 1n nb b for all 1,n

(ii)1

,nnb

(iii) lim 0,nn

b

(iv)1 nnc

and

1 n nnb c

.

If T satisfies condition (A), then the sequence {xn} converges strongly to afixed point of T.

139

Proof. It follows from Theorem 2.1 that

liminf 0.n nnx Tx

Since T satisfies condition (A), we have

liminf , 0.nnf d x FT

From the property of f, it follows that

lim inf , 0.nnd x F T

It follows from Lemma 2.1 that , 0nd x F T as n . Now, we can take an

infinite subsequence {jnx }of{xn}and a sequence jp F T such that

2j

jn jx p . Set 2

1exp 1nn

M k

and write 1j j ln n for some 1.l

Then we have

1jn jx p

jn l jx p

21 1j jn l n l jk x p

21 11 1

j jn l n l jk x p

21 1exp 1

j jn l n l jk x p

1

2

0

exp 1j j

l

n m n jm

k x p

2 j

M . (2.21)

It follows from (2.21) that

1j jp p 1 1 1j jj n n jp x x p

1

12 2j j

M

1

2 12 j

M

. (2.22)

Hence {pj} is a Cauchy sequence. Assume that jp p as j . Then p F T

since F(T) is closed, which implies that jx p as j . This completes the proof.

Remark 2.1. We note that, if T:KK is completely continuous, then it must be

140

demicompact [8], and if T is continuous and demicompact, it must satisfy condition(A) [4,9]. In view of this observation, Theorem 2.2 improves the correspondingresult of Liu [3] in the following aspects:(i) K may be not necessarily compact or bounded,(ii) T may be not uniformly Holder continous.Remark 2.2. Our results improve and generalize the corresponding results of[6,7,8,10,11,12] and many others from the existing literature.Remark 2.3. Our results also extend the corresponding results of Cho et al. [1] tothe case of more general class of -continuous asymptotically quasi-nonexpansivemappings.

REFERENCES[1] Y.J. Cho, G.T. Guo and H.Y. Zhou, Approximating fixed points of asymptotically quasi-

nonexpansive mappings by the iterative sequences with errors, Proceedings DynamicalSystems and Applications, Antalya, Turkey, July 5-10, 2004, 262-272.

[2] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically non-expansivemappings, Proc. Amer. Math. Soc. 35 No.1 (1972), 171-174.

[3] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mapping with anerror member of uniformly convex Banach space, J. Math. Anal. Appl., 266 (2002), 468-471

[4] Z. Opial, Weak convergence of successive approximations for nonexpansive mapping,Bull. Amer. Math. Soc., 73 (1967), 591-597.

[5] X. Qin, Y. Su and M. Shang, Strong convergence for three classes of uniformly equi-continuous and asymptotically quasi-nonexpansive mapping, J. Korean Math. Soc., 45No.1 (2008), 29-40.

[6] B.E. Rhoades, Fixed point iteration for certain nonlinear mappings, J. Math.Anal. Appl.,183 (1994), 118-120.

[7] J. Schu, Weak and strong convergence to fixed point of asymptotically nonexpansivemappings, Bull. Austral. Math. Soc., 43 No.1 (1991), 153-159.

[8] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings,J. Math. Anal. Appl. 158 (1991), 407-413.

[9] H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings,Proc. Amer. Math. Soc., 44 (1974), 375-380.

[10] K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by theIshikawa iteration process. J. Math. Anal. Appl., 178 (1993), 301-308.

[11] K.K. Tan and H.K. Xu, Fixed point iteration process for asymptotically nonexpansivemappings, Proc. Amer. Math. Soc., 122 (1994), 733-739.

[12] B.L. Xu and M.A. Noor, Fixed point iterations for asymptotically non-expansive mappingsin Banach spaces, J. Math. Anal. Appl., 267 (2002), No.2, 444-453.

141


IMPACT OF SEXUAL MATURATION ON THE TRANSMISSIONDYNAMICS OF HIV INFECTION IN HETEROGENEOUS COMMUNITY

: A MODEL AND ITS QUALITATIVE ANALYSISBy

Abha Teguria

Department of Mathematics, Government M.L.B. Postgraduate GirlsAutonomous College, Bhopal-462008, Madhya Pradesh, India

Manindra Kumar Srivastava

Department of Mathematics, School of Management Sciences, TechnicalCampus, Lucknow-226016, Uttar Pradesh, India


Anil Rajput

Department of Mathematics, Sadhu Basbani Postgraduate College,Bhopal-462003, Madhya Pradesh, India

(Received : July 10, 2010)

ABSTRACTIn this paper, we develop a model to study the spread of HIV infection, which

can cause Acquired Immunodificiency Syndrome (AIDS), through Vertical andhorizontal transmissions and introduce the concept of sexually immature andmature individuals by considering maturation rates in heterosexual communityand partially analyzed. We obtain equilibrium points of the system at two states(Disease-free and Endemic). We investigate the criteria for existence of endemicsteady state of the system. We determine local and global dynamics of these steadystates of the system and conclude.2010 Mathematics Subject Classification : Primary 92B05; Secondary 92D30Keywords: Vertical and Horizontal Transmissions, Heterosexual Community,Asymptotic Stability, Epidemiological Parameters.

1. Introduction. Today's, Acquired Immunodeficiency Syndrome (AIDS),has shown a very high degree of prevalence in populations all over the world,which is caused by Human Immunodeficiency Virus (HIV). The nature of humaninteractions, the uncertainties in the current estimates of epidemiologicalparameters and the lack of enough reliable data make it extremely difficult tounderstand the dynamics of the virus transmission without the frame worksprovided by mathematical modelling. The study of mathematical modelling is alsohelpful in determining the demographic and economic impact of the epidemic which

142

is turn help us to develop reasonable scientifically and socially sound investigationplans in order to reduce the spread of the infection.

In recent decades, several mathematical modelling studies have beenconducted to describe the transmission dynamics of HIV infection for homogeneousand heterogeneous populations, Anderson et al. ([1][2]), Bailey [5], Knox [16],Pickering et al. [21], May and Anderson ([17][20]), Grant et al. [12], Hethcote [13],Anderson et al. [3], May [18] Castillo-Chavez, Cooke, Huang and Levin ([7],[8],[9]),Sun [23], Chen [10] and Hethcote [13].

In particular, Anderson et al. [1] described some preliminary attempts touse mathematical models for transmission of HIV in a homosexual community.May and Anderson [17] presented simple HIV transmission models to help clarifythe effects of various factors on the overall pattern of AIDS epidemic. Blythe andAnderson (1988) considered HIV transmission models with four forms for thedistribution of incubation period by assuming that the infectious period is equalto the incubation period. Castillo-Chavez et al. [7] analyzed a model where themean rate of acquisition of new partners depends on the size of the sexually activepopulation. Most of the above mentioned models consider only one population butHIV transmission takes place in the population that are heterogeneous in a varietyof ways and this aspect should be taken in modelling HIV. Knox [16], Colgete et al.[11], Jacquez et al. [14], Koopman et al. [15].

2. Model Formulation. Let us consider a heterosexual community of sizeP with uniform promiscuous behaviour and taking only heterosexual encountersand assume that the birth and death rates are same, making community size to bea constant. We have assumed that infection passes in the considering populationthrough the member of one male or female class to the other female or male classrespectivley. The infection can also be transmitted vertically to the offspring ofinfected mother.

Let any instant of time t this considering community be subdivided into sixclasses of S1(t) mature male susceptibles, I1(t) mature male infective having HIVinfection, S2(t) mature female susceptibles, I2(t) mature female infective havingHIV infection, X(t) immature susceptibles and Y(t) immature infectives havingHIV infection. The susceptibles become infected with transmssion efficiency k andimmature susceptibles and infectives being sexually matured at rates m and m'respectively.

This leads to the following system of ordinary differential equations

11 2 1,

dSkS I mX bS

dt 1

1 2 1' ' ,DI

kS I m Y b Idt

143

22 1 21 ,

DSkS I mX bS

dt 2

2 1 21 ' ' ,dI

kS I m Y b Idt

' 1 ,dX

bS pb I mX mXdt

' ' 1 ' ,dY

qb I m Y m Ydt

(2.1)

with initial data

1 10 2 20 1 10 2 20 00 0, 0 0, 0 0, 0 0, 0 0,S S S S I I I I X X

0 1 2 1 20 0, , , ,1 ,0 1Y Y P S I X Y S S S I I I p q (2.2)

whereand (1–)= The proportions of maleand female hosts, who converts fromimmature class to mature classrespectively.b and b' are Birth rtes of immaturesusceptible and infectives respectively.P= The fraction of newborn offspring ofinfective parents, who are susceptible atbirth.q= The fraction of newborn offspringsof infective parents, who are infective atbirth.b and b' are also Death rate of maturesusceptibles and infectives respectively.The above mathematical modelling canbe understood by Fig.1.

3. Equilibrium Points ofModel. The equilibrium points of themodel can be derived from the followingset of equations

* * * *1 2 1 0kS I mX bS ,

* * * *1 2 1' ' 0kS I m Y b I ,

* * * *2 1 21 0kS I mX bS ,

* * * *2 1 21 ' ' 0kS I m Y b I ,

144

* * * * * *1 2 1 2' 1 0b S S pb I I mX mX ,

* * * *1 2' ' 1 ' 0qb I I m Y m Y ,

* * * * * *1 2 1 2S S I I X Y p . (3.1)

Disease Free Equilibrium Point * * * * * *0 1 1 2 2, , , , ,E S I S I X Y .

* * *1 2 0,I I Y

* *1 2

1, , ,

mPmP bPS S X

m b m b m b

Endemic Equilibrium Point * * * * * *1 1 1 2 2, , , , ,E S I S I X Y .

22 2 1 3*

11

42

a a a aS

a

,

*1 0;S if * * *'

' ' ,km

ka mX m Y bb Yq

24 4 1 5*

21

42

a a a aS

a

*2 0;S if * * *'

1 ' ' ,km

k mX m Y bb Yq

** 8 11

1

,2

a bSI

a

*1 0,I if * * *

1' ,a mX m Y bS

** 8 22

1'

a bSI

b

,

*2 0,I if * * *

21 ' .mX m Y bS

Now we investigate a criteria for endemic steady state E1 to exist. Here weneed that the system of equations (3.1) should have a positive solution.From the set of equation (3.1), we get

145

* *1 2 ,X b b Y (3.2)

which implies*

1X b when * 0Y and *1 2/Y b b when * 0X .

We can also obtain *

2* ,dX

bdY

*

* 0,dXdY

if ' ' ' 'm qb b pb m .

Hence, we find that X* is decreasing function of Y*

Also, form the set of equations (3.1) we get

1/ 22* * * * *

10 12 14 13 15 10 14 12 15 11 102 2 'a X ka a a Y a a X a ka Y a b a a X

2* * *10 14 12 15 10 161 1 ' ,a X a ka Y a b a a X (3.3)

which implies

4 3 2* * * *22 23 24 25 26 0,a X a X a X a X a when * 0.Y (3.4)

By Discarte's rule of sign the equation (3.4) has at least one positive root under

condition 2 ' 1.bb Let us denote that positive root by Q1.

Hence, *1X Q when * 0Y .

Further, when * 0X , we get

4 3 2* * * *27 28 29 30 31 0a Y a Y a Y a Y a . (3.5)

By Discarte's rules of sign, the equation (3.5) also has at least one positive rootunder condition 2bb'>1. Let us denote that positive root by Q2.

Hence *2Y Q when * 0X .

We can also obtain

3533 33 3514 12 12 14 131/2 1/ 2 1/2 1/ 2*

32 34 32 34*

3533 11 1610 10 11/ 2 1/ 2 1/2 1/ 2

32 34 32 34

12

.1 1

'2

aa a aa ka ka a a

a a a adXdY aa a a

a a b aa a a a

146

*

* 0dXdY

, if 3533

141/ 2 1/232 34

1 aaa

a a

33 35

12 14 131/ 2 1/ 232 34

2a a

ka a aa a

,

33 3512 14 131/ 2 1/ 2

32 34

2a a

ka a aa a

and

3533 11 161/ 2 1/ 2 1/ 2 1/ 232 34 32 34

1 10 ' 1

2aa a a

ba a a a

or if

3533 33 3514 12 14 131/ 2 1/ 2 1/ 2 1/ 2

32 34 32 34

12

aa a aa ka a a

a a a a

,

33 3512 14 131/ 2 1/ 2

32 34

2a a

ka a aa a

and

3533 11 161/ 2 1/ 2 1/ 2 1/ 232 34 32 34

1 1' 1

2aa a a

ba a a a

.

Hence, we find that X* isdecreasing function of Y*. Formabove, we see that the two isoclinesgiven by (3.2) and (3.3) will intersect

provided that 1 1Q b and 1

22

bQ

b .

Hence, with the aboveconsiderations, equation (3.2)represents X* as decreasing from b1and equation (3.3) represents X* asdecreasing from Q1.

Therefore, two isoclines mustintersect provided that intersection

value *10 X Q and *

1 20 /Y b b ,

then endemic equilibrium point E1Fig-2

147

exists in the positive Y*–X* plane, shown in Fig. 2.

Note : The values of constants ai (i=1,2,...,35) and bi (i=1,2) are given in theAppendix.

4. Qualitative Analysis. Now to determine local and global dynamics ofsteady states E0 and E1 of system (2.1) we use, Lyapunov's Second Method. Weanalyze local dynamic of above steady states (E0 and E1) and therefore findsufficient conditions under which E0

and E1 are locally asymptotically stable inthe form of the following Theorems 4.1 and 4.2 respectively:Theorem 4.1. Let the following inequalities hold

*1' ' / 2b kS m (4.1a)

*2 1b kS m (4.1b)

*2' 1 ' / 2b kS m (4.1c)

'm b pb (4.1d)

' 'm qb . (4.1e)

Then E0 is locally asymptotically stable.Proof. Using the following positive definite function in the linearized form ofsystem (2.1),

2 2 2 2 2 21 1 1 2 2 3 3 4 4 5 5 6 / 2V n A n A n A n A n A n , (4.2)

Where, Ai are arbitrary positive constants (i=1,2,...,6), a it can be checked that the

derivative of V1 with respect to t under the conditions (4.1) is negative definite.Hence in view of theory of stability, E0 is locally asympotically stable.Theorem 4.2. Steady state E1 of the system (2.1) is locally asymptotically stableif

* *1 2' ' / 2b kS kI m , (4.3a)

* *1 2 1 /2kI b kS m , (4.3b)

* *2 1' 1 ' / 2b kS kI m , (4.3c)

'm b pb , (4.3d)

' 'm qb , (4.3e)

and satisfied

148

Proof. Similar to Theorem 4.1.Now to show that steady states E0 and E1 of system (2.1) are globally

asymptotically stable, we consider a region of attraction for the system (2.1) inthe form of followingLemma 1. Consider the set

1 1 1 2 2 1 1 1 1 2 2, , , , , : 0 ,0 ,0 ,m m mR S I S I X Y S S P I I P S S P

2 20 ,0 ,0mI I P x Y

Which is a region of attraction for all solutions initially in the positive orthant,

where 1 2 1 2m m m mS S I I are positive constants.

Now we obtain here the conditions for asymptotic stabiity of positive steadystates E0

and E1 of system (2.1) in non-linear (global) case in the form of followingTheorem 4.3 . The steady state E0 of system (2.1) is non-linearly (globally)asmptotically stable in a region R1 given by Lemma 1 where the followingconditions are satisfied :

40 2B , (4.5a)

' ' / 2b kP m , (4.5b)

2 1 / 2mb kS m , (4.5c)

' 1 ' / 2b kP m , (4.5d)

'm b pb , (4.5e)

' 'm qb , (4.5f)

where B4 is arbitrary positive constant.Proof. Consider the following positive definite function about E0

2 2 2 2 2 22 1 1 2 2 3 3 4 4 5 5 6 / 2.V u B u B u B u B u B u (4.6)

HereBi are arbitrary positive constants (i=1,2,...,6).

Take perturbations in * * * * * *0 1 1 2 2, , , , ,E S I S I X Y as 1 2 3 4 5, , , ,u t u t u t u t u t and

6u t respectively and putting

* *1 1 1 1 1 2, ,S S u t I I u t

* *2 2 3 2 2 4, ,S S u t I I u t

149

* *5 6,X X u t Y Y u t ,

in the system (2.1), we get

*11 1 1 4 5

dubu k S u u mu

dt , (4.7a)

*22 1 1 4 6' '

dub u k S u u m u

dt , (4.7b)

*32 3 2 3 51

duk S u u bu mu

dt , (4.7c)

*42 3 2 4 6' 1 '

duk S u u b u m u

dt , (4.7d)

51 2 3 4 5' '

dubu pb u bu pb u mu

dt , (4.7e)

62 4 6' ' '

duqb u qb u m u

dt . (4.7f)

Differentiating V2 with respect to t along the solution of (2.1) and usingLemma 1, we get

2 2 2 2 2 221 1 1 2 3 3 4 4 5 5 6 1 1 4 4 1 5' ' ' m

dVbu b B u bB u b B u mB u m B u kS u u m bB u u

dt

1 2 4 1 2 6 2 2 2 3 2 4 3 5 3 2 4' 1mkPB u u m B u u kS B u u mB bB u u kPB u u

3 4 6 4 2 5 4 4 5 5 2 6 5 4 61 ' ' ' ' 'm B u u pb B u u pb B u u qb B u u qb B u u . (4.8)

Applying the inequality 2 2 / 2ab a b and making algebraic

manipulation, we derive

2 2 2 2 2 2211 1 12 2 13 3 14 4 15 5 16 6

dVB u B u B u B u B u B u

dt , (4.9)

where

11 1 4 / 2,mB b kS m bB

12 1 1 2 2 1 3 4 5' ' ' ' / 2mB b B kPB kS B m B kPB pb B qb B ,

13 2 2 2 2 41 / 2mB bB kS B mB bB ,

150

14 3 1 1 3 3 5 4' 1 ' ' ' / 2mB b B kS kPB kPB m B qb B pb B ,

15 4 2 41 2 ' / 2,B mB m mB b pb B

16 5 1 3 5' ' 1 ' 2 ' / 2.B m B m B m B qb B

From (4.9), it can be shown that, 2dVdt

is negative definite under following conditions

1 4 / 2mb kS m bB , (4.10a)

1 1 2 2 3 1 5' ' ' 'mb B kPB kS B kPB m B qb B pb B , (4.10b)

2 2 2 2 41 /2,mbB kS B mB bB (4.10c)

3 1 1 3 3 5 4' 1 ' ' ' / 2,mb B kS kPB kPB m B qb B pb B (4.10d)

4 4 4 22 2 ' 1 /2,mB m bB pb B mB (4.10e)

5 1 3 5' ' 1 ' 2 ' / 2.m B m B m B qb B (4.10f)

The above sufficient conditions for 2dVdt

to be negative dfinite, may be further

manipulated to get the following simplified conditions:In (4.10a) choosing B4 as

40 2B , ...(4.11a)

and in (4.10b) choosing B1 as

2 2 3 4 51

' '' ' / 2

mkS B kPB pb B qb BB

b kP m ,

the condition (4.10b) reduces to

' ' / 2b kP m . (4.11b)

In (4.10c) choosing B2 as

4

22

/ 2 ,

1 / 2m

bBB

b kS m

the condition (4.10c) reduces to

2 1 / 2mb kS m . (4.11c)

151

In (4.10d) choosing B3 as

1 1 4 5

3

' ',

' 1 ' / 2mkS kPB pb B qb B

Bb kP m

the condition (4.10d) reduces to

' 1 ' / 2b kP m . (4.11d)

In (4.10e) choosing B4 as

24

1 /2 ,

'

m mBB

m b pb

the condition (4.10e) reduces to

'm b pb . (4.11e)

In (4.10f) choosing B5 as

1 3

5

' 1 / 2,

2 ' 2 '

m B BB

m qb

the condition (4.10f) reduces to

' 'm qb . (4.11f)

Hence, disease free steady state E0 of system (2.1) is non-linearly (Globally)asymptotically stable in the region R1 under the conditions given by (4.11), provigTheorem 4.3.Similarly we also determine that steady state E1 of system (2.1) is also globallyasymptotically stable in a region R1

given by Lemma 1 under the followingconditions :

*2 1 1 4 / 2mkI b m kPD kS bD , (4.12a)

*2' ' / 2b kS kP m , (4.12b)

*1 2 1 / 2mkI b kS m , (4.12c)

*1' 1 ' / 2b kI kp m , (4.12d)

'm b pb , (4.12e)

' 'm qb , (4.12f)

where D1 and D4 are arbitrary positive constants.5. Conclusion. In this model, disease free and endemic steady states, have

152

been obtained, which are shown to be both linearly (locally) and non-linearly(globally) asymptotically stable under the conditions involving disease relatedparameters.

From the qualitative analysis of the disease free steady state E0, it may beconcluded that the infection will die out eventually in the underlying populationand only mature and immature population of susceptible males and females willexist. From the qualitative analysis of the endemic steady state E1, it may beconcluded that the infection will remain always in the population provided thefollowing disease related parameters satisfy

* * * * * *1 2' , 1 ' ,2 ' 1, ' and ' 'mX m Y bS mX m Y bS bb m b pb m qb .

APPENDIX

* * * *1 2 10 14 12 15 3 6 7, , ,a kb a a X a ka Y a a a X a Y

* *

4 10 14 12 151 1a a X a ka Y a , * *5 6 71a a X a Y

* *6 7 8 10 11 12 13 14

'', ', ' , , 4 , , 4 ', ',

ma mb a m a mX m Y a km a b a a kpb a km

q

15 16 17 14 13 12 18 14 12 19 14 12', 4 1 , 2 , 1 , ,a bb a b a a a ka a a ka a a ka

20 14 16 14 15 12 15 21 14 11 14 15 12 152 1 2 , 2 2 ,a a a a a ka a a a a a a ka a

4 24 2 4 2 2 3 2

22 10 10 101 2 1 2 1 ,a a a a

3 23 3 2 223 10 15 10 10 151 1 5 / 2 4 1 8 ,a a a a a a

4 24 4 2 2

24 10 10 151 4 2 1 5a a a a ,

22 2 2 3 2 225 10 15 10 15 26 15 152 1 60 , 4 1 4 ,a a a a a a a a

4 4 4 2 2 4 2 227 17 18 19 18 19 17 18 192 2 ,a a a a a a a a a

2 2 2 2 2 2 2 228 18 20 19 21 17 15 19 20 18 21 17 20 21 17 15 18 192 2 8 2 4a a a a a a a a a a a a a a a a a a

4 4 2 2 2 2 2 2 2 2 2 2 2 2 229 18 19 18 15 19 15 17 15 19 15 20 21 15 18 15 18 192 2 24 2 4a a a a a a a a a a a a a a a a a a

2 217 15 20 21 17 154 2 ,a a a a a a

153

2 2 2 2 3 2 2 2 330 18 15 19 15 17 15 21 15 20 15 15 20 21 17 152 2 32 2 4 8a a a a a a a a a a a a a a a a ,

2 2 2 *31 15 15 32 33 10 114 1 4 , ' ,a a a a a b a a X * *

33 10 14 12 15,a a X a ka Y a

2 * * *34 35 10 16 35 10 14 12 15' , 1 1 ,a a b a a X a a X a ka Y a

1 2

1 '/ ' '/ ',

'/qb m pb b mPb

b bm b m b qb b

.

ACKNOWLEDGEMENTSWe are grateful to Dr. O.P. Misra, Reader School of Mathematics and Allied

Sciences, Jiwaji University Gwalior, M.P., India for his constructive commentsand valuable suggestions during the preparation of this paper.

REFERENCES[1] R.M. Anderson, G.F. Medley, R.M. May and A.M. Johnson, IMA JI. Math. Appl. Med.

Biol., 3 (1986), 229-263.[2] R.M. Anderson, S.P. Blythe, G.F. Medley and A.M. Johnson, Lancet, 10 (1987).[3] R.M. Anderson, R.M. May and A.R. McLean, Natures, 332, 6161 (1988), 228-234.[4] R.M. Anderson, J.R. Statist. Soc., A. 151 (1989), 66-93.[5] N.T.J. Bailey, B. In Blum, M. Jorgansen eds., Medinfo Elsevier, 86 (1986), 741-744.[6] S. Busengerg and K. Cooke, Vertically Transmitted Diseases Model Dynamics: M/S

Springer-Verlag, New York, 1992[7] C. Castillo-Chavez, K. Cooke, W. Huang and S.A. Levin, On the Role of Long Periods of

Infectiousness in the Dynamics of Acquired Immunodeficiency Syndrome (AIDS),Mathematical Approaches to Problems in Resource Management and Epidemiology (eds.C. Castillo-Chavez, S.A. Levin and Shoemaker), Lecture Notes in Biomethematics, No.81, Springer Verlag, New York, (1989), 177-189.

[8] C. Castillo-Chavez, K. Cooke, W. Huang and S.A. Levin, On the Role of long incubationperiods in the Dynamics of Acquired Immunodeficiency Syndrome (AIDS), Part 1, SinglePopulation Models, J. Math. Bio., 27 (1989), 373-398.

[9] C. Castillo-Chavez, K. Cooke, W. Huang and S.A. Levin, On the Role of Long IncubationPeriods in the Dynamics of Acquired Immunodeficiency Syndrome (AIDS), Part 2, MultipleGroup Models, Mathematical and Statistical Approaches of AIDS Epidemiology (Eds. C.Castillo-Chavez), Lecture Notes in Biomathematics, NO. 83, Springer Verlag, New York(1989), 200-217.

[10] J. Chen, Proc. Int. Conf. Math. Biol., May 1997, China.[11] S.A. Colgate, E.A. Stanley, J.M. Hyman, et al. LA-UR-8733412 (Los Alamos Technical

Report), 1989.[12] R. Grant, J. Willey and W. Winklestein, J. Inf. Dis., 156 (1987), 189-193.[13] H.W. Hethcote, Future Trends in AIDS, Her Majesty's Stationery Office, London (1987),

35-40.

154

[14] J.A. Jecquez, C.P. Simon and J. Koopmam, Mathematical and Statistical Approaches toAIDS Epidemiology, C. Castillo-Chavez (ed) Lecture Notes in Biomethematics, 83,Springer-Verlag, 1989.

[15] J.S. Koopman, S.P. Simon and J.A. Jacquez, Mathematical and Statistical Approachesto AIDS Epidemiology, C. Castillo-Chavez (ed) Lecture Notes in Biomethematics, 83,Springer-Verlag, 1989.

[16] E.G. Knox, Eur. J. Epidemiol., 2 (1986), 165-177.[17] R.M. May and R.M. Anderson, Nature, 326 (1987), 137-142.[18] R.M. May, Nature, 33 (1988), 665-666.[19] R.M. May, R.M. Anderson and A.R. McLean, In Proceedings International Symposium

in Mathematical Approaches to Ecological and Environmental Problem Solving (eds. C.Castillo-Chavez, S.A. Levin and C. Shoemaker), Lecture Notes in Biomathematics,Springer-Verlag. New York, 1988.

[20] R.M. May and R.M. Anderson, Biomathematics, Simon, A. Levin, Thomas, G. Hallam &Louis, J. Gross (eds.), Applied Mathematical Ecology, 18, Springer-Verlag, 1989.

[21] J. Pickering, J.A. Wiley, N.S. Padian, et al., Math. Modelling,7 (1986), 661-698.[22] M.R.M. Roi, and S. Ahmad, Theory of Ordinary Differential Equations with Applications

in Biology and Engineering, East-West Press Pvt. Ltd. New Delhi, 1999.[23] J. Sun, Biometrics, 51, (1995), 1096-1104.

155


ON UNIT OF LENGTH MEASUREMENT IN THE INDUS-SARASWATICIVILIZATION AND SPEED OF LIGHT IN THE VEDIC LITERATURE

ByM.R. Goyal

122C, South City, Gurgaon-122007, HaryanaE-mail:[email protected]

(Received : October 18, 2010; Revised June 6, 2011)

ABSTRACTAn exact value of the unit of length measurement used in Indus-Saraswati

Civilization, has been determined from the precise scale discovered by ErnestMackay in the 1930-31 season excavation at Mohenjodaro and further correlatedwith the present day units of measurement. It has been calculated and shown thatthe speed of light as given in the Vedic literture, when referenced with this erstwhileunit of length measurement (used in Indus-Saraswati Civilization), works out tobe precisely equal to the speed of light as per modern measurements. The presentpaper is a step-wise process followed to unveil this equality.2010 Mathematics Subject Classification : Primary 01AXX Secondary 01A32Keywords: Unit of length measurement, Indus-saraswati civilization, speed oflight in vedic literature.

1. The Precise Scale. In his 1930-31 season at Mohanjo-daro, ErnestMackay discovered a broken piece of shell bearing 8 divisions of precisely 6.7056mmeach, with a dot and circle five graduations apart, which suggests a decimal system.However, attempts by Mackay, to relate such a unit to dimensions in Mohanjo-daro, were not very successful (Michel Danino [2]) and thus were abandoned.

2. Units of Length in Chanakya's Arthashastra. Chanakya was thepolitical of the legendary monarch Chandragupta Maurya of 4th century BC. Hewas aman learned in may disciplines and wrote the famous treatise on economicscalled the Arthashastra-meaning the books of money). In Arthashastra, Chanakyamentions two types of Dhanushas as units for measuring length and distances.One is the ordinary Dhanusha, consisting of 96 Angulas, and the other Dhanushais mentioned as Garhpatya Dhanusha and consists of 108 Angulas. Chanakya alsomentions many other units including a Dhanurgraha, which consists of 4 Angulasand a Yojana (In Sanskrit-English dictionary, by Moniere William, a Yojana hasbeen defined as a mesure of distance=4 Kosas or about 9 miles. Apart from theMoniere William dictionary, several other books/sources also give the Yojana aseqal to about 9 miles.), as consisting of 8000 Dhanushas.

156

3. Decoding the Mohan-jo-daro Scale. If we keep 10 divisions of theMohanjo-daro scale as equal to a Dhanurgraha or 4 Angulas, the precise length ofan Angula works out to be 16.764mm.A Dhanusha of 96 Angulas=96×16.764mm=1.609344m (1)andA Dhanusha of 108 Angulas= 108×16.764mm =1.810512meters. (2)

1 Yojana=8000 Dhanushas (of 108 Angulas each) (3)Thus

1 Yojana=8000 ×1.810512m=14.484096km. (4)Further

14.484096km=9 miles, (exactly!). (5)Also

1000 Dhanushas of 96 Angulas each=1mile (6)Interestingly, when we look into the history of mile, we find that the word

mile is derived from mile, which means a thousand. This points to universaladaptation of ancient units of length.

4. Corroboration from Other SourcesThe Indus Inch: The Indus civilisation unit of length, widely known as IndusInch was 1.32 Inches which is exactly equal to 2 Angulas of 16.76mm each.Mohenjo-daro's Great Bath: The height of the corbelled drain forming the outletof Mohenjo-daro's Great Bath [6, pp. 133-142] is about 1.8m, which is equal to aDhanusha of 108 Angulas of 16.764mm each.Standard Street-Widths. Kalibangam, a city in the Indus-Saraswati Civilization(in Rajasthan) had street widths [8] of 1.8m, 3.6m, 5.4m and 7.2m i.e. built to thestandard dimensions being equal to 1 Dhanusha, 2 Dhanushas, 3 Dhanushas and 4Dhanushas respectively. Such widths are found at other sites also. Bigger streetsof Banawali [8] another town in Indus-Saraswati Civilization (in Haryana) measure5.4m i.e. they were built with the unit of 3 Dhanushas.Taj Mahal. A Persian manuscript "Shah Jahan Nama" contains a very particulardescription of three principal buildings of Agra-the Taj Mahal, Moti Masjid andJamah Masjid. In the "Shah Jahan Nama", the dimensions of these three buildingsare given in Gaz. These dimensions were got measured by col. J.A. Hodgson inDecember, 1825, in feet and inches. The various (28) dimensions, in feet and inches,as well as in Gaz, are given by Hodgson in his article [5], in Table A. He has alsogiven Table B (25 dimensions) excluding 3 dimensions which he thought were notvery dependable. The weighted mean length of a Gaz works out to 31.70 inches,(80.52cm) from Table A and 31.66 inches=80.42cm from Table B. The average ofthe two values in 31.68 inches=80.47cm. It is pertinent to mention here thatBarraud [1, pp. 108-109, 258-259] in The complete Taj Mahal and the River front

157

Columns of Agra, has taken a Gaz as equal to 80.5cm which is very nearly equal to80.47cm as worked out above. Taking a Dhanusha of 96 Angulas to be equal to 2Gaz, the length of an Angula works out to 16.764mm. It shows that a Gaz of 48Indus-Saraswati Angulas was being used, even in the days of Shah Jahan's rulei.e. 17th Century A.D.The Gudea's Rule. The Gudea's rule (2175 B.C.) preserved in the Louvre showsintervals in Sumerian Shusi of 0.66 inches, which is exactly equal to the Indus-Saraswati Angula of 16.764mm.Mayan Units of Measurement. Drewitt [3] and Drucker [4] made a study of theancient city of Teotihuacan, belonging to Mayan Civilization, in Mexico, andhypothesised a unit of 80.5cm, which is very nearly equal to the Indian Guz of 48Angulas=80.47cm.Temple Wall-Engravings. Nearer home, two engravings on a wall of the templeat Tiruputtkali (12th Century A.D.) near Kanchipuram, show two scales [7] onemeasuring 7.24 metres in length, with marketing dividing the scale into 4 equalparts, and the second one measuring 5.69 metres in length and marking dividingthe scale into 4 equal parts. It may be observed that each division of the first scaleis precisely equal to a Dhanusha of 108 Angulas of 16.764mm each. Interestingly,the second scale is precisely equal to times Dhanusha i.e. equal to thecircumference of a circle with one Dhanusha as its Diameter.

It is interesting to note here that Mackay reports [9] at Mohenjo-daro, alane and a doorway having both a width of 1.42m, which is precisely equal to onedivision of the second scale at the Tiruputtkali Temple, indicating that both thescales were prevalent in Indus-Saraswati Civilization as well as in South India.

It proves beyond doubt that the Units of measurement as derived from theprecise scale found at Mohenjo-daro were prevalent not only in the Indus-SaraswatiCivilization, but also in South India and in the ancient Sumerian, Egyption andMayan Civilizations.

5. Speed of Light in Vedic Literature. In the commentary on Rig-Veda.Mandal 1, Sukta 50, Mantra 4, which is in praise of the Sun god, Sayanacharya(14th Century AD) writes:

rFkk p Le;Zrs---

;kstukuka lglza }s }s 'krs }s p ;kstusA

,dsu fufe"kk/ksZu Øeek.k ueksLrq rsAAMeaning... "It is remembered that...Salutations to Thee (the Sun) who approacheth (at a speed of) 2202 yojanas

in a nimishardha (half nimisha)."Clearly it is the Speed of light (or sunrays) that is mentioned in the Shloka.

158

This shloka is attributed [10, pp. 67] to the son of Kanva Maharshi (4000 B.C.).Bhatta Bhaskara (10th Century) mentions [11] this shlok in his commentary onthe Taitreya Brahamana.

To put it in mathematical terms, as stated by the above shlok, the speed oflight would be :

Speed of Light= 2202 Yojanas/Nimishardha (7)We have already calculated the value of Yojana in modern unit of km in the

previous step (Equation 4), but we still do not know, what the Nimishardhatranslates into. Let us dig into another Vedic text the Vishnu Puran to find howthe erstwhile units of time can be related to the modern units of time.

In the Vishnu Puran (Book 1, Chapter 3, Shloka 8,9), it is stated that:15 Nimishas = 1 Kashtha30 Kashthas = 1 Kala30 Kalas = 1 Mahurta

30 Mahurtas = 1 day and night (vgksjk=ke~)Thus

one day and night = 405,000 Nimishas=810,000 Nimishardhas. (8)(Literal meaning of Nimishardha being half of Nimisha).In Surya Sidhant (Chapter 1, Shloka 12), it is mentioned that 60 Nadis

constitute one Sidereal Day and Night (uk{k=ke~ vgksjk=ke~). It is also well known that1 Mahurta=2 Ghatis or 2 Nadis. It is clear from this that in Astronomicalcalculations, the sidereal day was taken as the unit of time. A sidereal day is thetime taken by the stellar constellations to complete one revolution around theEarth. A sidereal day is equal to 23 hours, 56 minutes and 4.1 seconds or equivalentsec (Wikipedia [12]).

ThusNimishardha=86164.1/810000 sec=0.1063754sec (9)

The speed of light as given in the Vedic Literature therefor comes out to be;

52202*14.4840962.998 *10 /sec

0.1063754 seckm

km . (10)

which is precisely equal to the speed of light as per modern measurements.6. Conclusions. The calculations and references int he article endeavor to

establish the vast reach of the Indus Valley scholars, not only academically, butalso geographically. Thousands of years before the modern scientists rediscoveredthe speed of light; our ancestors knew of the exact same value and referred to it asan exalted property of the Sun-God to salute Him. It is also heartening to knowthat the modern-age concept of globalization and sharing of ideas between cultures

159

was also an established way of life in their era, as proven by the shared units ofmeasurement found across far-flung cultures. When and how these ancientcivilization lost their influence is still a mystery, but for now we can celebrate theknowledge that :1. The Speed of Light as given in the Vedic literature, is precisely equal to theSpeed of Light as per modern measurements.2. The basic unit of length measurement in the Indus-Saraswati Civilizationwas an Angul of 16.764mm. This unit was used not only in the Indus-SaraswatiCivilization, but also in South India, and other ancient world Civilizationsincluding Sumerian, Egyptian and Mayan Civilizations.

REFERENCES[1] R.A. Barraud, The Complete Taj Mahal and The Riverfront Gardens of Agra (ed. Koch

E.) Thames and Hudson. London, 2006.[2] Danino, Michel, The Lost River-On the Trail of Saraswati, Penguin Books, 2010.[3] Drewitt, Measurement Units and Building Areas at Teotihuacan, 1987[4] R. David Drucker, Precolumbian Mesoamerican Measurment Systems: Unit Standards

for Length, 1977.[5] Col. J.A. Hodgson, Memoir on the Length of the Illahi, Guz, Journal of the Royal Asiastic

Society of Great Britain and Ireland., 1840.[6] Jansen, Michael Mohenjo-daro: architecture et urbanisme, Les cités oubliées de lIndus:

Archéologie due Pakistan, 1988.[7] James Heitzman and S. Rajagopal, Urban Geography and Land Measurement in the

Twelfth Century: The Case of Kanchipuram. The Indian Economic and Social HistoryReview, SAGE, 2004.

[8] B.B. Lal, The Earlist Civilization of South Asia, Aryan Books International, New Delhi,1997.

[9] E.J.H. Mackey, Further Excavations at Mohenjo-daro, Govt. of India. Vol. I republishedMunshiram Manoharlal, 1998.

[10] Pride of India, Published by Sanskrit Bharati, 2006.[11] Subhash Kak, The speed of Light and Puranic Cosmology, 2001.[12] Wikipedia Article on Sidereal Time, http://en.wikipedia.org/wiki/Sidereal_time.

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