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In investigation of boundary-value problems for certain partial differentialequations arising in applied mathematics, we often need to study the solutionof system of partial differential equations satisfied by hypergeometricfunctions and find explicit linearly independent solutions for the system. Herewe choose the Exton function $K_{2}$ among his 21 functions to show how to findthe linearly independent solutions of partial differential equations satisfiedby this function $K_{2}$. Based upon the classical derivative and integraloperators we introduce a new operational images for hypergeometric function$K_{2}$. By means of these operational images a number of finite series anddecomposition formulas are then fund.

Linear independent solutions and operational representations for hypergeometric functions of four variables