Confluent hypergeometric functions of many variables and their application to the finding of fundamental solutions of the generalized Helmholtz equation with singular coefficients

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 of one variable has stimulated the development of the corresponding theory for functions of two and more variables. In the theory of hypergeometric functions, an increase in the 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 in terms of an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitates the process of studying the properties of multidimensional functions. Confluent hypergeometric functions in all respects, 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, give integral representations, and establish decomposition formulas. An important application of confluent functions has been found. It turns out that 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, the order of the singularity of the fundamental solutions of the above elliptic equation is determined.

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