Linear Independent Solutions and Operational Representations for Hypergeometric Functions of Four Variables

In investigation of boundary-value problems for certain partial differential equations arising in applied mathematics, we often need to study the solution of system of partial differential equations satisfied by hypergeometric functions and find explicit linearly independent solutions for the system. Here we choose the Exton function K2 among his 21 functions to show how to find the linearly independent solutions of partial differential equations satisfied by this function K2. Based upon the classical derivative and integral operators, we introduce a new operational images for hypergeometric function K2. By means of these operational images, a number of finite series and decomposition formulas are then found.