Linearly Independent Solutions and Integral Representations for Certain Quadruple Hypergeometric Function

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:msubsup> <a:mrow> <a:mi>X</a:mi> </a:mrow> <a:mrow> <a:mn>84</a:mn> </a:mrow> <a:mrow> <a:mfenced open="(" close=")"> <a:mrow> <a:mn>4</a:mn> </a:mrow> </a:mfenced> </a:mrow> </a:msubsup> </a:math> , 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M2"> <e:msubsup> <e:mrow> <e:mi>X</e:mi> </e:mrow> <e:mrow> <e:mn>84</e:mn> </e:mrow> <e:mrow> <e:mfenced open="(" close=")"> <e:mrow> <e:mn>4</e:mn> </e:mrow> </e:mfenced> </e:mrow> </e:msubsup> </e:math> .

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