Generating Relations Involving Hypergeometric Function by Means of Integral Operators

It should be noticed that the exponential function, which is a special case of the generalized hypergeometric function pFq and p = q = 0, appears in many different situations; for instance, in conformal mapping theory [151], in automorphic function theory [151], in the theory of representation of Lie algebras, in Physics and in the theory of differential equations [87]. The leading example of partly bilateral and partly unilateral generating function in terms of exponential function is doubtless the result of Exton ([38]; p. 147(3)] Where and are the classical Laguerre polynomials ([121; p. 200(1)). Pathan and Yasmeen [109] modified Exton's result (1.1) by defining and So that all factorials of negative integers occurring in this definition have meaning. Thus we may rewrite equation (1.1) in the form This result has attracted a great deal of interest by several researchers including (for example) Pathan and Yasmeen ([107], [108] and [109]). Goyal and Gupta [52], Srivastava et al. ([142] and [143]). Gupta et al [60], Kamarujjama et al. [79], and Pathan and Subuhi [116]. Works on Exton‘s result inspired us to use integral operators to obtain more generating functions, which are partly bilateral and partly unilateral. On account of many properties of generating functions which are partly bilateral and partly unilateral, an increasing, number of such problems and properties are now capable of being elegantly represented by their use. A number of such, generating functions are obtained in this paper. explored and some generating functions are obtained by making use of results of Exton [38] and Pathan and Yasmeen [109]. Further, a number of multiple seri4s of hypergeometric functions are obtained in Section 1.3.

GENERATING FUNCTIONS:

If we define the integral operator  by Then rewriting the results ([130] p. 36(6)), ([45]; p.192(50), with y = 1), ([45]; p.193 (51), with y = 1) in terms of this operator, we have the results And Starting from the result (2.1) with z replaced , we have Now using the results (1.3), (2.1) and (2.2), we can establish the following result Further letting one has the following closure relation which for  = µ, can be written as, Again taking in (2.1) for we have Now using the above result, together with the results (2.3) and (1.3), we are led to For equation (2.8) reduce to

The method of derivation of the generating functions involves the following results. A result of Pathan ([106]; p. 52(5)) Where is a generalized hypergeometric function of (n+1) variable ([106]; p. 51(1)), Where and by analytic continuation, none of the quantities c, e1, e2, … , en are zero or a negative integer. The result of Pathan and Yasmeen ([108]; p.5(3.1)) Expressing Laguerre polynomial in terms of confluent hypergeometric function using (1.2) and further using the relation between and the Whittaker‘s function of the first kind

We see that the Laguerre polynomial is related to the Whittaker function by the equation Now, replacing x1, x2 ad x3 by x1u, x2u and x3u, respectively in (3.3) and multiplying both the sides by Further using (3.4) to replace each of the Laguerre polynomials, and finally integrating with respect to u from zero to infinity, we arrive at Now using result (3.1), we get, after some simplifications respectively and also by a and b respectively to get

1. and redplacing 3-a by a, We get Where is Lauricella function [95] of n variables, defined as follows Further for n = 3, equation (3.8) reduces to 2. For n = 3 equivalently for x4 = x5 = … = xn = 0, equation (3.7) gives us Where is hyper geometric function of four variables considered by Pathan [8] and is defined as follows

[1] Goyal and Gupta (1989) ―Certain double generating functions for the generalized Hyper Geometric functions‖ Buttetin Institute of Mathematics, Academia Sinica 17, no. 4. [2] Pathan M.A. and Yasmeen (1968) ―A note on 4 new generating relation for a generalize Hypergeometric function‖ Jour, Math. Phy Science. [3] Pathan M.A. & Khan, Subuhi ―Generating relations involving Hypergeometric functions by means of Integral operations‖ , preprint. [4] Srivastava H.M., Pathan M.A. & Kamamujama M. (1988) : ―Some unified Presentations of the Generalized voight functions. Common Appl‖. Anal. 2 [5] Hussain, M.A., Kamrujjama, M. and Aftab, F. (1997): On partly bilateral and partly unilateral generating functions, Soochow JI of mathematical, China.

Research Scholar, Department of Mathematics, V.K.S. University, Ara 802301, Bihar, India