TWO DIMENSIONAL LAPLACE TRANSFORMS OF GENERALIZED HYPERGEOMETRIC FUNCTIONS

The object of this paper is to obtain new operational relations between the original and the image functions that involve generalized hypergeometric G-functions.

The integral equation (p,q) pq f f exp(-px-qy)f(x,y) dy dx, Re(p,q) > 0 (1.1) O0 represents the classical Laplace transform of two variables and the functions (p,q) and f(x,y) related by (1.1), are said to be operationally related to each other.(p,q) is called the image and f(x,y) the original.
Symbolically we can write (p,q) f(x,y) or f(x,y) (p,q), (1.2) and the symbol is called "operational".
Meijer's G-function [5] is defined by a Mellin-Barnes type integral where m, n, u, v are integers with v ) I; 0 < n < u; 0 4 m < v, the parameters a.3 and b.3 are such that no poles of r(bj-s); j 1,2 ,m coincides with any pole of F(1-ak+s) k 1,2 n.Thus (ak-bj) is not a positive integer.The path L goes from -i to +i so that all poles of integrand must be simple and those of F(bj-s); j 1,2,...,m lie on one side of the contour L and those of F(1-ak+s) k 1,2,...,n must lie on the other side.The integrand converges if u+v (2(re+n) and larg z < (m + nu -v).For sake of brevity au denotes al,a 2, ,a u.
In the present paper, we propose to establish a couple of formulae for calculating Laplace transform pairs of two dimensious that involve Meijer's G- function.
From (2.1) and (2.2), we propose to prove the following relations.
PROOF: The generalized Stieltjes transform of a G-function is given by (see [6], p. 237) a I-G, a 8/-0 Gm+l'n+l (a8 On writing pq for B and multiplying both the sides of (2.5) by p q, we have Now interpreting with the help of the known result ([4], result (2.83), p.
Evaluating the left-hand side integral of (2.The following result will be used in the proof of (2.2).
On specializing the parameters, the G-functlon can be reduced to MacRobert's Efunction, generalized hypergeometric function and other higher transcendental functions.Therefore, the results (2.1) and (2.2) leads to many new operational relations not listed in [4,9,2,3] and other literature.
3(a).lamed image functions expressed in ter of the G-function.
The following results are obtained by using the known results from [7] on pages 226-334.