Lauricella hypergeometric function $F_{A}^{(n)}$ with applications to the solving Dirichlet problem for three-dimensional degenerate elliptic equation

Abstract

 In this paper, hypergeometric function of Lauricella  $F_{A}^{(n)}$   has  been investigated.  The new properties of which are established and applied to the solution of the Dirichlet problem for the three-dimensional degenerate elliptic equation. Fundamental solutions of the named equation are expressed through the Lauricella hypergeometric function in three variables and an explicit solution of the Dirichlet problem in the first octant is written out through the Appell hypergeometric function  $F_2$. A limit theorem for calculating the value of a function of many variables is proved, and formulas for their transformation are established. These results are used to determine the order of singularity of fundamental solutions and to prove the truth of the solution to the Dirichlet problem. The uniqueness of the solution to the Dirichlet problem is proved by the extremum principle for elliptic equations.