Reduction formulas for hypergeometric functions and their application to the comparison of problems solutions for the Laplace equation and singular elliptic equations

The main goal of this work is to prove a number of reduction formulas for the first Lauricella function, i.e. to determine the set of those numerical parameters for which the hypergeometric function is expressed in elementary functions. Currently, tables of partial values for Appell functions of two variables are known; however, these formulas are insufficient for applications, and therefore, it is necessary to establish more general relations for hypergeometric functions whose number of variables exceeds 2. In the first part of the work, we will prove new reduction formulas for the Lauricella function in three and more variables, generalizing previously known results. The second part of the paper is devoted to applying the obtained reduction formulas to the study of the properties of solutions to the Dirichlet and Neumann problems for the Laplace equation and a singular elliptic equation. The main result of this study is that, using reduction formulas for hypergeometric function, it is proven that the solutions to the Dirichlet and Neumann problems for elliptic equations with vanishing singular coefficients coincide with the solutions to similar problems for the Laplace equation