Fundamental Solutions of the Generalized Helmholtz Equation with Several Singular Coefficients and Confluent Hypergeometric Functions of Many Variables

Abstract

An investigation of applied problems related to heat conduction and dynamics, electromagnetic oscillations and aerodynamics, quantum mechanics and potential theory leads to the study of various hypergeometric functions. The great success of the theory of hypergeometric functions in one variable has stimulated the development of corresponding theory in two and more variables. In the theory of hypergeometric functions, an increase in a number of variables will always be accompanied by a complication in the study of the function of several variables. Therefore, the decomposition formulas that allow us to represent the hypergeometric function of several variables through an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitate the process of studying the properties of multidimensional functions. In the literature, hypergeometric functions are divided into two types: complete and confluent. In all respects, confluent hypergeometric functions including the decomposition formulas, have been little studied in comparison with other types of hypergeometric functions, especially when the dimension of the variables exceeds two. In this paper we define a new class of confluent hypergeometric functions of several variables, study their properties and determine the system of hypergeometric equations that these functions satisfy, because all fundamental solutions of the generalized Helmholtz equation with singular coefficients are written out through one new introduced confluent hypergeometric function of several variables. Using the decomposition formulas which are established here the order of the singularity of the found fundamental solutions of the elliptic equation which mentioned above is determined.