A three-variable analogue of the Humbert function with applications to solving Dirichlet problem for a singular Helmholtz equation

In this paper we introduce a new confluent hypergeometric function of three variables, list its elementary properties, and construct a system of partial differential equations that the new function satisfies. We study the behavior of the function and establish an asymptotic formula for a large values of arguments. The results obtained are applied to solving the Dirichlet problem for the three-dimensional Helmholtz equation with three singular coefficients in the first infinite octant. The uniqueness of the solution of the Dirichlet problem in an infinite domain is proved using the extremum principle for elliptic equations. Thanks to the established properties of the confluent hypergeometric function of three variables, the unique solution to the posed boundary value problem is written out in explicit form.

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