Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variables

TL;DR

This paper introduces a new class of confluent hypergeometric functions of many variables, explores their properties, and applies them to explicitly construct fundamental solutions of a generalized Helmholtz equation with multiple singular coefficients.

Contribution

The paper develops a novel class of confluent hypergeometric functions of many variables and demonstrates their use in explicitly formulating fundamental solutions for a generalized Helmholtz equation.

Findings

Fundamental solutions are expressed via the new confluent hypergeometric functions.

The properties and differential equations satisfied by these functions are characterized.

The singularity order of solutions is determined using the expansion formula.

Abstract

In this paper, we introduce a new class of confluent hypergeometric functions of many variables, study their properties, and determine a system of partial differential equations that this function satisfies. It turns out that all the fundamental solutions of the generalized Helmholtz equation with several singular coefficients are written out through the newly introduced confluent hypergeometric function. Using the expansion formula established here for the confluent function, the order of the singularity of the fundamental solutions of the elliptic equation under this consideration is determined.