Linearly Independent Solutions and Integral Representations for Certain Quadruple Hypergeometric Function

Abstract

Recently, hypergeometric functions of four variables are investigated by Bin-Saad and Younis. In this manuscript, our goal is to initiate a new quadruple hypergeometric function denoted by $X_{84}^{\left(4\right)}$ , and then, we ensure the existence of solutions of systems of partial differential equations for this function. We also establish some integral representations involving the quadruple hypergeometric function $X_{84}^{\left(4\right)}$ .