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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 4 , 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 4 .

Linearly Independent Solutions and Integral Representations for Certain Quadruple Hypergeometric Function