ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 16, Number 2, 2012 Available online at www.math.ut.ee/acta/ Bilinear generating relations for a family of -polynomials and generalized basic hypergeometric functions S. D. Purohit, V. K. Vyas, and R. K. Yadav Abstract. In this paper, we derive a bilinear -generating function involving basic analogue of Fox s H-function and a general class of -hypergeometric polynomials. Applications of the main results are also illustrated. 1. Introduction and preliminaries For a, C the -shifted factorial (see 2) is defined by { 1 ; n 0 (a ; ) n (1 a) (1 a ) (1 a n 1 ) ; n N, and its natural extension is (1.1) (a ; ) α (a ; ) (a α ; ), α C, < 1. (1.2) The definition (1.1) remains meaningful for n as a convergent infinite product (a ; ) (1 a j ). (1.3) j0 The -analogue of the power (binomial) function (x ± y) n (cf. Ernst 1) is given by n (x ± y) (n) (x ± y) n x n ( y/x; ) n x n n k k(k 1)/2 (±y/x) k, (1.4) Received March 20, 2012. 2010 Mathematics Subject Classification. 33D45, 33D60, 33D90. Key words and phrases. -Generating function, basic analogue of Fox s H-function, -polynomials. 191
192 S. D. PUROHIT, V. K. VYAS, AND R. K. YADAV where the -binomial coefficient is defined as α ( α ; ) k ( k α ) k k(k 1)/2 (k N, α R). (1.5) (; ) k It satisfies n k k (; ) k. (1.6) For a bounded seuence A n of real or complex numbers, let f(x) A n x n be a power series in x, (see, for instance, 1, page 502, euation (3.18)), then we have f (x ± y) A n x n ( y/x; ) n. (1.7) The -gamma function (cf. 2) is defined by And it satisfies Γ (x) (; ) ( x ; ) (1 ) 1 x (x C, x / {0, 1, 2, ). (1.8) Γ (x + 1) 1 x 1 Γ (x). (1.9) In terms of a bounded complex seuence {S k,, the family of general class of basic (or -) polynomials f n,m (x; ) (cf. Srivastava and Agarwal 9) is defined as f n,m (x; ) where m is a positive integer. n/m n mk S k, x k (n N), (1.10) With the appropriate choice of the seuence {S k,, the -polynomial family f n,m (x; ) yields a number of known -polynomials as its special cases. These include, the -Hermite polynomials, the -Laguerre polynomials, the -Jacobi polynomials, the Wall polynomials, the -Konhauser polynomials and several others. Following Saxena, Modi and Kalla 8, the basic analogue of the Fox s H-function is defined as H M,N P,Q x; (a, α) 1 θ(s; ) x s d s, (1.11)
BILINEAR GENERATING RELATIONS 193 where and θ(s; ) { { M G( b j β j N s ) G( 1 a j+α j s ) π j1 { Q { G( 1 b j+β j s ) jm+1 P jn+1 j1 G( a j α j s ) G( 1 s ) sin π s (1.12) { 1 G( a ) (1 ) a+n 1 ( a ; ). (1.13) Also 0 M Q, 0 N P, α i s and β j s are all positive integers. The contour C is a line parallel to R(ω s) 0 with indentations if necessary, in such a manner that all the poles of G( b j β j s ), 1 j M are to the right, and those of G( 1 a j+α j s ), 1 j N to the left of C. For large values of s, the integral converges if Rs log(x) log sin πs< 0 on the contour C, i.e. if { arg(x) w 2 w1 1 log x < π, where 0< <1, log w (w 1 +iw 2 ), w 1 and w 2 being real. Further, if we set α i β j 1, i and j in (1.11), we obtain the basic analogue of Meijer s G-function due to Saxena, Modi and Kalla 8: G M,N P,Q x; a 1, a 2,, a P b 1, b 2,, b Q 1 θ (s; ) x s d s, (1.14) where θ (s; ) { Q { { M N G( bj s ) G( 1 aj+s ) π j1 G( 1 bj+s ) jm+1 { P j1 jn+1 G( a j s ) G( 1 s ) sin π s (1.15) A detailed account of Meijer s G-function, Fox s H-function and various functions expressible in terms of Fox s H-function can be found in the research monographs due to Mathai and Saxena 4, 5, Mathai, Saxena and Haubold 6 and Srivastava, Gupta and Goyal 10. Further, the basic functions of one variable (elementary and hypergeometric) expressible in terms of the functions G (.) can be found in the works of Yadav and Purohit 12 and 13..
194 S. D. PUROHIT, V. K. VYAS, AND R. K. YADAV 2. The -generating relations In this section, we shall derive certain bilinear -generating relations involving basic analogue of Fox s H-function and a general class of -hypergeometric polynomials. Theorem 1. Let {S k, be an arbitrary bounded seuence, let M, N, P, Q be positive integers such that 0 M Q, 0 N P, let h > 0, and let m be an arbitrary positive integer. Then the following bilinear -generating relation holds: f n,m (ρ x; ) H M,N+1 y; (1 λ n, h), (a, α) t n H M,N+1 1 S k, (1 t) (λ) (; ) mk y (1 t λ+mk ) (h) ; (ρ x t m ) k (1 t λ ) (mk) (1 λ mk, h), (a, α) where t < 1, 0 < < 1, and ρ and λ are arbitrary numbers., (2.1) Proof. Denoting, for convenience, the left-hand side of (2.1) by L and using the contour integral representation (1.11) for the basic analogue of Fox s H-function and the definition (1.10) for the general class of -polynomials f n,m (ρ x; ), we get L 1 n/m n S 2π i mk k, (ρ x) k { θ(s; )G( λ+n+hs ) y s t n d s. C Changing the order of summations and integration, we obtain L 1 θ(s; ) n/m G( λ+n+hs ) n mk S k, (ρ x) k t n y s d s, (2.2) where θ(s; ) is given by (1.12). Using of the relation for -gamma function, namely G( a ) Γ (a) (1 ) a 1 (; ), (2.3) we obtain L 1 θ(s; ) Γ (λ + hs) (1 ) λ+hs 1 (; )
n/m BILINEAR GENERATING RELATIONS 195 ( λ+hs ; ) n n mk S k, (ρ x) k t n y s d s. Again, changing the order of summations and making use of the series rearrangement relation (cf. Srivastawa and Manocha 11) we obtain n/m B(k, n) B(k, n + mk), (2.4) L 1 θ(s; ) Γ (λ + hs) (1 ) λ+hs 1 (; ) (ρ x t m ) k ( λ+hs ; ) n+mk S k, t n y s d s. (2.5) (; ) mk Summing the inner series with the help of the -binomial theorem (see 2), namely we find that 1Φ 0 (a; ; ; z) (az; ) (z; ), z < 1, 0 < < 1, (2.6) L 1 θ(s; ) Γ (λ + hs) (1 ) λ+hs 1 (; ) ( λ+hs ; ) mk (ρ x t m ) k S k, y s d s. (2.7) (t; ) λ+hs+mk (; ) mk Now by interchanging the order of contour integral and summation, and using the -identities (see 2), namely and we obtain (a; ) n L 1 (t; ) λ (a; ) n+k (a; ) n (a n ; ) k (2.8) Γ(a + n)(1 )n Γ(a) (ρ x t m ) k (t λ ; ) mk (; ) mk S k, (n > 0), (2.9) 1 θ(s; ) Γ (λ + hs + mk) (1 ) λ+hs+mk 1 (t λ+mk y s d s. (2.10) ; ) hs (; ) The desired result follows by interpreting the contour integral of (2.10) in light of the definition (1.11) and the notation (2.3). This completes the proof of Theorem 1.
196 S. D. PUROHIT, V. K. VYAS, AND R. K. YADAV Observe that, if we set the bounded seuence S k, 1 and take ρ 0, then for the family of -polynomials one has f n,m (ρ x; ) 1, and thus in view of the right-hand side of (2.1) for k 0, we obtain the following theorem. Theorem 2. Let M, N, P, Q be positive integers satisfying 0 M Q, 0 N P. Let h > 0, let λ be an arbitrary number, and let m be an arbitrary positive integer. Then the -generating relation for the basic analogue of Fox s H-function is given by H M,N+1 y; (1 λ n, h), (a, α) H M,N+1 y (1 t λ ) (h) ; where t < 1 and 0 < < 1. t n (1 λ, h), (a, α) 3. Concluding observations and remarks 1 (1 t) (λ), (2.11) In this section, we consider some conseuences of the results derived in previous section. If we set α i β j 1 for all i and j, m h 1, and take (1.14) into account, then Theorems 1 and 2 yield Corollaries 1 and 2 below, respectively. Corollary 1. Let {S k, be an arbitrary bounded seuence and let M, N, P, Q be positive integers satisfying 0 M Q, 0 N P. Then the following bilinear generating relation for the function G (.) holds: f n,1 (ρ x; ) G M,N+1 y; S k, (; ) k 1 λ n, a 1,, a P t n 1 b 1,, b Q (1 t) (λ) 1 λ k, a 1,, a P, b 1,, b Q (ρ x t) k y (1 t λ GM,N+1 )(k) (1 t λ+k ) ; where t < 1, 0 < < 1 and λ is an arbitrary number. (3.1) Corollary 2. Let M, N, P, Q be positive integers satisfying 0 M Q, 0 N P and let λ be an arbitrary number. Then the -generating relation for the basic analogue of Meijer s G-function is given by G M,N+1 y; 1 λ n, a 1,, a P b 1,, b Q t n 1 (1 t) (λ)
BILINEAR GENERATING RELATIONS 197 G M,N+1 y (1 t λ ) ; 1 λ, a 1,, a P, (3.2) b 1,, b Q where t < 1 and 0 < < 1. Further, it is interesting to observe that in view of the following limiting cases: lim Γ ( a ; ) n (a) Γ(a) and lim 1 1 (1 ) n (a) n, (3.3) where (a) n a(a + 1) (a + n 1), (3.4) the -generating relation (2.1) of Theorem 1 provides the -extension of the known result due to Raina 7, page 301, euation (2.1). By assigning suitable special values to the seuence {S k,, our main result (Theorem 1) can be applied to derive certain bilinear -generating relations for the product of orthogonal -polynomials and the basic analogue of Fox s H-function. To illustrate this, we consider the following example. Setting m 1 and we find from (1.10) that S k, ( 1)k k(k 1) (α; ) n (α; ) k, (3.5) f n,1 (x; ) L (α) n (x; ), where L (α) n (x; ) denotes the -Laguerre polynomial defined by (cf. 9) L (α) n (x; ) (α; ) n 1Φ 1 n ; x n α ;. (3.6) Thus in view of the above relations, Theorem 1 yeilds the -generating relation involving -Laguerre polynomial and the basic Fox s H-function as below: L (α) n (ρ x; ) H M,N+1 y; (1 λ n, h), (a, α) t n (α; ) n ( 1) k k(k 1) (ρ x t) k (1 t) (λ) (; ) k (α; ) k (1 t λ ) (k) H M,N+1 y (1 t λ+k ) (h) ; (1 λ k, h), (a, α). (3.7) Again, if we set m 1 and S k, (α; ) n(αβ n+1 ; ) k ( 1) k k(k+1)/2 nk (α; ) k, (3.8)
198 S. D. PUROHIT, V. K. VYAS, AND R. K. YADAV we find from (1.10) that f n,1 (x; ) P n (α,β) (x; ), where P n (α,β) (x; ) denotes the -Jacobi polynomial defined by (cf. 9) P n (α,β) (x; ) (α; ) n 2Φ 1 n, αβ n+1 ;x α ;. (3.9) Then, Theorem 1 provides the -generating relation involving -Jacobi polynomial and the basic Fox s H-function, namely P n (α,β) (ρ x; ) H M,N+1 y; (1 λ n, h), (a, α) t n (α; ) n (1 t) (λ) H M,N+1 y (1 t λ+k ) (h) ; (αβ n+1 ; ) k ( 1) k k(k+1)/2 nk (; ) k (α; ) k (1 λ k, h), (a, α) (ρ x t) k (1 t λ ) (k). (3.10) A detailed account of various hypergeometric orthogonal -polynomials can be found in the research monograph by Koekoek, Lesky and Swarttouw 3 and in 9. It is worth mentioning that the definitions of -Laguerre and -Jacobi polynomials given by the euations (3.6) and (3.9), respectively, are slightly different from those given in the seminal work 3. Therefore, one can derive similar type of results by taking into consideration the definitions of the -polynomials given in 3. We conclude with the remark that by suitably assigning values to the seuence {S k,, the -generating relation (2.1) being of general nature, will lead to several generating relations for the product of orthogonal - polynomials and the basic analogue of the Fox s H-functions. Acknowledgments The authors would like to thank the anonymous referee for the critical review and comments on the manuscript, leading to the present form of the paper.
BILINEAR GENERATING RELATIONS 199 References 1 T. Ernst, A method for -calculus, J. Nonlinear Math. Phys. 10(4) (2003), 487 525. 2 G. Gasper and M. Rahman, Basic Hypergeometric Series (2nd edition), Cambridge University Press, Cambridge, 2004. 3 R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their -analogues, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. 4 A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lecture Series in Mathematics, Vol. 348, Springer-Verlag, Berlin New York, 1973. 5 A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, Halsted Press John Willey and Sons, New York London Sidney, 1978. 6 A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-function. Theory and Applications, Springer, New York, 2010. 7 R. K. Raina, A formal extension of certain generating functions, Proc. Nat. Acad. Sci. India Sec. A 46(IV) (1976), 300 304. 8 R. K. Saxena, G. C. Modi, and S. L. Kalla, A basic analogue of Fox s H-function, Rev. Tec. Ing. Univ. Zulia, 6 (1983), 139 143. 9 H. M. Srivastava and A. K. Agarwal, Generating functions for a class of -polynomials, Ann. Mat. Pura Appl. 154 (1989), 99 109. 10 H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publications, New Delhi, 1982. 11 H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press John Wiley and Sons, New York, 1984. 12 R. K. Yadav and S. D. Purohit, On applications of Weyl fractional -integral operator to generalized basic hypergeometric functions, Kyungpook Math. J., 46 (2006), 235 245. 13 R. K. Yadav, S. D. Purohit, and S. L. Kalla, On generalized Weyl fractional -integral operator involving generalized basic hypergeometric functions, Fract. Calc. Appl. Anal. 11(2) (2008), 129 142. (S.D. Purohit) Department of Basic Sciences (Mathematics), College of Technology and Engineering, M.P. University of Agriculture and Technology, Udaipur-313001, India E-mail address: sunil a purohit@yahoo.com (V.K. Vyas, R.K. Yadav) Department of Mathematics and Statistics, J.N. Vyas University, Jodhpur-342005, India E-mail address: rkmdyadav@gmail.com