An investigation of applied problems related to heat conduction and dynamics, electromagnetic oscillations and aerodynamics, quantum mechanics and potential theory leads to the study of various hypergeometric functions. The great success of the theory of hypergeometric functions in one variable has stimulated the development of corresponding theory in two and more variables. In the theory of hypergeometric functions, an increase in a number of variables will always be accompanied by a complication in the study of the function of several variables. Therefore, the decomposition formulas that allow us to represent the hypergeometric function of several variables through an infinite sum of products of several hypergeometric functions in one variable are very important, and this, in turn, facilitate the process of studying the properties of multidimensional functions. In the literature, hypergeometric functions are divided into two types: complete and confluent. In all respects, confluent hypergeometric functions including the decomposition formulas, have been little studied in comparison with other types of hypergeometric functions, especially when the dimension of the variables exceeds two. In this paper we define a new class of confluent hypergeometric functions of several variables, study their properties and determine the system of hypergeometric equations that these functions satisfy, because all fundamental solutions of the generalized Helmholtz equation with singular coefficients are written out through one new introduced confluent hypergeometric function of several variables. Using the decomposition formulas which are established here the order of the singularity of the found fundamental solutions of the elliptic equation which mentioned above is determined.

The main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients. These fundamental solutions are directly connected with multiple hypergeometric functions and the decomposition formula is required for their investigation which would express the multivariable hypergeometric function in terms of products of several simpler hypergeometric functions involving fewer variables. In this paper, such a formula is proved instead of a previously existing recurrence formula.The order of singularity and other properties of the fundamental solutions that are necessary for solving boundary value problems for degenerate second-order elliptic equations are determined

In this work the definition of the order of the double hypergeometric series is given, the new higher-order Kampé de Fériet functions are introduced into the consideration, their properties are studied. The results obtained are applied to finding an explicit regular solution to the Cauchy problem for a degenerate hyperbolic equation of the second kind.

The present work is devoted to study the boundary value problems for a multidimensional Laplace equation with the Dirichlet conditions on the lateral faces of the ball’s hyperoctant and on the sphere’s hyperoctant. The main result is establishment of necessary and sufficient conditions for unique solvability of the boundary value problems. An energy integral method and Green’s function method are used in the proof of the main result. All solutions are found in an explicit forms, which contain singular solution of Laplace equation.

In this article, a special class of generalized solutions of the Cauchy problem for the hyperbolic equation with two lines of degeneracy of the second kind is introduced and the properties of this class of generalized solutions is studied. Using the properties of the Pochhammer symbol and of the Appell hypergeometric function of two variables, the solution of the Cauchy problem in an explicit and convenient form is found.