5 Gaper and Rahman [5], ha given the following relation between qcontigou baic hypergeometric function (5) (6) (7) and (8) The following reult are due

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1 International Journal of Theoretical & Applied Science, 5(): 4 (03) ISSN No (Print): ISSN No (Online): Some Recurrence for Generalzed Hypergeometric Function Awadheh Kumar Pandey*, Ramakant Bhardwaj**, Kamal Wadwa*** and NS Thakur* Department of Mathematic, *Patel Intitute of Technology, Bhopal, (MP) ISSN No (Print) : **Truba Intitute of Technology, Bhopal, (MP) ***Govt Narmda PG College, Hohangabad, (MP) ISSN No (Online)(Received : November, 0, Accepted 05 January, 03) ABSTRACT: In the preent paper our reult i the qextenion of the known reult due to Galu e and Kalla [4] Which are define for Generalized hypergeometric function S+FS( ) in termof an iterated qintegral involving Gau hypergeometric function F( ) By uing the relation between qcontiguou hypergeometric function F( ), ome new & known recurrence relation for the generalized hypergeometric function of one variable are deduced in the line of Purohit [6] a a pecial cae I INTRODUCTION The generalized baic hypergeometric erie Gaper and Rahman [5] i given by r r (,, [ ; b,, b ; ) ; (b ); r ] () where for real or complex () i the qhifted factorial, r and are poitive integer, and variable, the numerator parameter denominator parameter b,, b,being any complex quantitie provided that bj, m 0,, ; j,,, If q <, the erie () converge abolutely for all converge abolutely if q > and if r and for <,, and the < if r + Thi erie alo Further, in term of the qgamma function, () can be expreed a ( ; q)n, n>0, (3) where the qgamma function (Gaper and Rahman [5]) i given by where 0,,, (4)

2 5 Gaper and Rahman [5], ha given the following relation between qcontigou baic hypergeometric function (5) (6) (7) and (8) The following reult are due to Galu e and Kalla [4], + where Re(b i) > 0, for all i 0,,,, In view of limit formulae where ( ( ) ) ( (9) and and (0) ), Due to Gaper and Rahman [5], the qanalogue of Euler integral repreentation 0 () The generalization of qanalogue of Euler integral repreentation due to Gaper and Rahman [5] i r r Purohit [6], ha given the iterated qintegral repreentaion + ()

3 where Re(b i) > 0, for all i 0,,, ; q <, 6 (3) and Bailey [], Exton [3], Galu e and Kalla [4], Gaper and Rahman [5] & Slater [8], ha given a wide range of application of the theory of generalized hypergeometric function of one and more variable in variou field of Mathematic, Phyic and Engineering Science, namelynumber theory, Partition theory, Combinatorial analyi, Lie theory, Fractional calculu, Integral tranform, Quantum theory etc In the preent article, Generalized hypergeometric function S+ S ( ) i expreed in term of an iterated qintegral involving the qgau hypergeometric function F( ) By uing qcontiguou relation for F( ), ome recurrence relation for the generalized hypergeometric function of one variable are obtained in the line of Purohit [6] a a pecial cae Here the mentioned technique i a qverion of the technique ued by Galu e and Kalla [4] II MAIN RESULT If Re(b i) > 0, for all i 0,,, and q <, then the iterated qintegral repreentaion of S+FS ( ) i given by + where, r > 0 and Proof: Uing equation (),the LHS of equation(0) can alo be written a S+ () repeating the proce in the righthand ide of (), we have S+ () (0)

4 7 again repeating the proce in the righthand ide of (), we have S+ 3 (3) On ucceive operation ( 4) time in the righthand ide of (3), we get S+ (4) which i the required reult III RECURRENCE RELATIONS Uing equation (5), we get On ubtituting value from relation (3) in the righthand ide of the equation (0), we have S+ (3)

5 8 (3) Again, on making ue of the reult (0), the above reult (3) lead to the following recurrence relation: S+ + + where Re( (33) ) > 0, for all i 0,,, and x < Uing equation (6), we get (34) On ubtituting value from equation (34) in the righthand ide of the equation (0), we have S+ (35)

6 9 Again, on making ue of the reult (0), the above reult (35) lead to the following recurrence relation: S+ + + where Re( (36) ) > 0, for all i 0,,, and x < Uing equation (7), we get (37) On ubtituting value from equation (37) in the righthand ide of the equation (0), we have S+ (38) Again, on making ue of the reult (0), the above reult (38) lead to the following recurrence relation: S+ + + (39)

7 Where, Re( 0 ) > 0, for all i 0,,, and x < Uing equation (8), we get (30) On ubtituting value from equation (30) in the righthand ide of the equation (0), we have S+ (3) Again, on making ue of the reult (0), the above reult (3) lead to the following recurrence relation: S+ + + where Re( (3) ) > 0, for all i 0,,, and x < We conclude with the remark that the reult deduced in the preent paper appear to be a new contribution to the theory of generalized hypergeometric erie Secondly, one can eaily obtain number of recurrence relation for the generalized hypergeometric function by the application of iterated qintegral repreentation for +FS( ) Remark: If we put r 0 in all the reult, we get the reult of [6]

8 ACKNWLEDGEMENNT Correponding Auther (AP) i thankful to all reviewer and Auther whoe reference are taken to prepare thi article REFERENCES [] Bailey W N, Generalized Hypergeometric Serie, Cambridge Univerity Pre, Cambridge, Second edition (964) [] Erd elyi A, Higher trancendental function, McGraw Hill, New York, vol (953) [3] Exton H, qhypergeometric Function and Application, Elli Horwood Limited, Halted Pre, John Wiley and Son, Chicheter, (983) [4] Galu e L and Kalla SL, Some recurrence relation for the generalized hypergeometric function of the Gau type, Fract Calc and Appl Anal (), (998), pp496 [5] Gaper G and Rahman M, Baic Hypergeometric Serie, Cambridge Univerity Pre, Cambridge, (990) [6] Purohit, SD, Some recurrence relation for the generalized baic hypergeometric function, Bulletin of Mathematical analyi and Application, Vol, Iue,(009), pp 9 [7] Rainville ED, Special function [8] Slater LJ, Generalized Hypergeometric Function, Cambridge Univerity Pre, Cambridge, London and New York, (966)

ON THE MELLIN TRANSFORM OF A PRODUCT OF HYPERGEOMETRIC FUNCTIONS

J. Autral. Math. Soc. Ser. B 40(998), 37 ON THE MELLIN TRANSFORM OF A PRODUCT OF HYPERGEOMETRIC FUNCTIONS ALLEN R. MILLER H. M. SRIVASTAVA (Received 7 June 996) Abtract We obtain repreentation for the