Generating Functions of Special Triple Hypergeometric Functions

Abstract In this paper we have obtained some generating relations involving Appell’s, Exton’s, Horn’s, Lauricella’s and Saran’s functions. We obtained these relation by using integral representation of Exton’s functions X11, X12, ..., X20 in terms of Laplace’s integral of several variables. Through the development of these new generating relations, we recover some Exton’s, Srivastva’s and Karlsson’s results.


Introduction
The unification of generating functions has great importance in connection with ideas and principles of special functions.In this directions some important steps has been made by researchers namely Singhal and Srivastava [13], Chaterjea [1,2] and Chongdar [3].In their study Desale and Qashash [4] have obtained a new general class of generating functions for the generalized modified Laguerre polynomials L (α) n (x) by group theoretic method.Also, they have introduced the bilateral generating function for the generalized modified Laguerre and Jacobi polynomials with the help of two linear partial differential operators.Further, they continued their study and they used the group theoretic method to obtain the proper and improper partial bilateral as well as trilateral generating functions.In this consent one may refer their sequel of papers [5,6].Now continuing the work in connection with class of generating functions, we extend our ideas to obtain the new generating relations that involves between Exton's functions and hyper geometric functions, in particularly Appell's functions, Lauricella's, Horn's, Saran's and Gaussian hyper geometric functions.We used integral form of Extonn's functions (in the form of Laplace integral) to obtain the new generating relations among Exton's and hypergeometric functions.
Exton in [9,10] gave integral representation of some hyper geometric functions of three variables out of which we referred here X 11 , X 12 , ..., X 20 1 and these relations are defined as follows (cf.[9,10]).
The integral representation of these Exton functions in terms of Laplace integral are given in the following section.

Integral Representation of Exton Functions
Here, we expressed Exton functions X 11 , X 12 , . . ., X 20 in terms of multiple integrals with the help of Laplace transform.Following are these integral representation of Exton functions. (2.10)

New Generating Relations
In this section, we used the integral representation of Exton functions to determine new generating functions which are listed in the form of following relations (3.1) to (3.18).

Proof of relation 3.1:
Let us denote the left hand side of (3.1) by A, using (2.1) by using (4.1) and (4.2), we get The function Ψ 2 which appears in above eqaution can be replaced by its series form and then interchanging the order of the summation and integral sign which is permissible here, we get •e (−t(1+k)) s α+n+2p+r−1 t β+n+2q+m−1 dsdt Now, we use sequentially (4.12), (4.6), (4.7) and (4.8); and then simplified by using series manipulation, which complete the proof of relation (3.1).
Remark 1.The proof of all remaining relations runs in the same way, considering the appropriate integral representation and Laplace transform during the proof.

Special Cases
Some generating relations, which are belived to be new, can be established in this section as the special cases of the results were obtained in previous section.

Conclusion
We used the Laplace integral representation of Exton's functions given by (1.1) to (1.10) to determine the new generating relations (3.1) to (3.18).Many of these results are the relations between Appell's functions F 1 , F 2 , F 3 ; Horn's functions H 3 , H 4 ; Saran's functions F K , F M , F N and Gaussian hypergeometric function 2 F 1 .One of the special case of (5.1) is the generating relation between Horn's and Exton's functions.The generating relation (5.2) is the relation between Gaussian hypergeometric function and Appell's functions.Also, we have seen in the section of special cases many of the results from section 3 reduces to Exton's functions for n = 0.