Certain Integrals Associated with Hypergeometric Functions of Four Variables

Abstract

The main objective of this paper is to present integral representations of Euler type and Laplace type for five new hypergeometric series of four variables.

Linearly Independent Solutions and Integral Representations for Certain Quadruple Hypergeometric Function

“…Here, we give eight integral representations of Euler type for X ð4Þ 84 whose kernel contains the Gaussian hypergeometric function 2 F 1 (see [16]), Appell function F 3 (see for details [16,25]), the Exton triple functions X 16 , X 17 , X 19 [26], Lauricella's function of three variables F N [16], and the quadruple functions X ð4Þ 4 , X ð4Þ 24 (see [20,21]):…”

“…Recently, various interesting hypergeometric functions in several variables have been investigated by many authors (see, e.g., [17][18][19][20][21][22][23][24]). In Section 2, we show how to find the linearly independent solutions of partial differential equations satisfied by the function X ð4Þ 84 .…”

Certain Generating Functions Involving Some Hypergeometric Series of Four Variables by Means of Operational Representations

“…It may be recalled the Laplace integral representations of the above functions, see e.g. [8,9,10,11] respectively, as below…”

“…Proof. To prove (21), for convenience and simplicity, by denoting the left-hand side of (21) with δ and using , , 4 , , ; , 1 2 , , ; , , , , , , , 2 1 , , , ; , , , ; , , , , , , , 3 1 4 2 2 2 1 1 2 2 3 1 1 1 ) 4 ( 4 2 4 3 2 1 2 3 2 1 1 2 X is quadruple hypergeometric series defined by Bin-Saad and Younis (see [9] and [7]). 1 3 1 2 3 2 1 3 0 0 4 1 1 3 3 2 3 2 3 2 3 1 3 1 1 2 2 7 1 2 1 , , ; , , ; , , 1 1 !…”

Certain integral representations involving hypergeometric functions in two variables

“…Choi et al [9] introduced certain integral representations for Srivastava's triple hypergeometric functions H A , H B and H C . Younis and Bin-Saad [18,19] establish several integral representations and operational relations involving quadruple hypergeometric functions X (4) i (i = 38, 40, 45, 48, 50). Younis and Nisar [20] introduce new integral representations of Euler-type for Exton's hypergeometric functions of four variables D 1 , D 2 , D 3 , D 4 and D 5 .…”