Integral Representations for Certain Quadruple Hypergeometric Series

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…”

“…In this section, we consider the symbolic images for the quadruple functions (1) - (10). Our results presented here are obtained with the help of the following operator formulas:…”

Some Integrals Connected with a New Quadruple Hypergeometric Series

“…More recently, Bin-Saad et al [2][3][4][5] introduced new hypergeometric series of four variables namely by X…”

“…10 and investigated their certain properties including integral representations, symbolic representations, generating functions, etc. Motivated largely by the aforementioned works of Bin-Saad et al [4] and [5], we defined further quadruple hypergeometric functions as follows:…”

“…Moreover, multiple hypergeometric functions can be used to solve physical and chemical problems in many areas of applied mathematics [1,14,19]. In the present study we aim to obtain certain integral representations of Euler-type and Laplace-type involving new quadruple hypergeometric series namely by X (4) i ( i = 11, 12,13,14,15). Recall the Gaussian hypergeometric function defined by [19] 2 The Appell series F 1 , F 2 and the Horn's series H 3 of two variables are defined as follows [19]:…”