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 among his 21 functions to show how to find the linearly independent solutions of partial differential equations satisfied by this function . Based upon the classical derivative and integral operators, we introduce a new operational images for hypergeometric function . By means of these operational images, a number of finite series and decomposition formulas are then found. 1. Introduction and Preliminaries Solutions of many applied problems involving thermal conductivity and dynamics, electromagnetic oscillation and aerodynamics, and quantum mechanics and potential theory are obtainable with the help of hypergeometric (higher and special or transcendent) functions [1–4]. Such kinds of functions are often referred to as special functions of mathematical physics. They mainly appear in the solution of partial differential equations which are dealt with harmonic analysis method (see [5]). In view of various applications, it is interesting in itself and seems to be very important to conduct a continuous research of multiple hypergeometric functions. For instance, in [6], a comprehensive list of hypergeometric functions of three variables as many as 205 is recorded, together with their regions of convergence. It is noted that Riemann functions and the fundamental solutions of the degenerate second-order partial differential equations are expressible by means of hypergeometric functions of several variables (see [7–19]). Therefore, in investigation of boundary-value problems for these partial differential equations, we need to study the solution of the system of hypergeometric functions and find explicit linearly independent solutions (see [12–17, 19]). Exton [20, pages 78-79] introduced 21 complete hypergeometric functions of four variables. In [21] Sharma and Parihar introduced 83 complete hypergeometric functions of four variables. It is remarked that, out of these 83 functions, the following 19 functions had already appeared in the work of Exton [20] in the different notations: Each quadruple hypergeometric function is of the form where is a certain sequence of complex parameters and there are twelve parameters in each function . Here, for example, we choose the Exton function among his twenty-one functions to find the