Differential Equations and Mathematical Physics

Confluent hypergeometric functions and their application to the solution of Dirichlet problem for the Helmholtz equation with three singular coefficients

Z. O. Arzikulov^a, A. Hasanov^bc, T. G. Ergashev<sup>bdc

a</sup> Fergana State Technical University, Fergana, 150107, Uzbekistan
^b V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science, Tashkent, 100174, Uzbekistan
^c Ghent University, Ghent, 9000, Belgium
^d Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Tashkent, 100000, Uzbekistan

Abstract: In the course of a series of studies spanning the fifty-year period from 1889 to 1939, all double hypergeometric series of the second order were systematically investigated. A significant contribution to the study of hypergeometric functions of two variables was made by Horn, who proposed their classification into two types: complete and confluent. Horn's final list comprised fourteen complete (non-confluent) functions of two variables and twenty distinct confluent functions, which represent limiting cases of the complete ones. In 1985, Srivastava and Karlsson completed the classification of all possible second-order complete hypergeometric functions of three variables, while a similar systematic classification for their confluent counterparts remains incomplete. Thus, the theory of confluent hypergeometric functions of three variables has not yet been fully developed, and the study of functions of four variables represents an area for future research.
This paper investigates certain confluent hypergeometric functions of three and four variables, establishing their new properties and applying them to the solution of the Dirichlet problem for the three-dimensional Helmholtz equation with three singular coefficients.
Fundamental solutions of the aforementioned Helmholtz equation are expressed in terms of a confluent hypergeometric function of four variables, while an explicit solution to the Dirichlet problem in the first octant is constructed using a function of three variables, which is derived as a trace of the four-variable confluent function. A theorem on the computation of limiting values of multivariate functions is proved, and transformation formulas for these functions are established. These results are employed to determine the singularity order of fundamental solutions and to validate the correctness of the solution to the Dirichlet problem.
The uniqueness of the solution to the Dirichlet problem is proved using the maximum principle for elliptic equations.

Keywords: multiple confluent hypergeometric function, PDE-systems of hypergeometric type, singular Helmholtz equation, fundamental solution, Dirichlet problem

UDC: 517.956

MSC: 33C70, 35J25, 35J05, 35A08

Received: February 24, 2025
Revised: May 2, 2025
Accepted: May 5, 2025
First online: August 22, 2025

DOI: 10.14498/vsgtu2156

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