Expansions of some double hypergeometric functions and their application to the solving boundary value problems.

In this paper, we will find expansion formulas for 5 hypergeometric functions of two variables from Horn's list, which includes 34 functions, and for 11 of them, expansion formulas were previously known according to works of Burchnall and Chaundy. To solve the problem, we introduce operators that generalize the well-known Burchnall-Chaundy's operators, study the properties of these new operators, and apply them to finding expansion formulas. At the end of the paper, to give an example, we will show the application of the expansion formula for one of the five functions to determine the order of the singularity at the origin of the fundamental solution used in solving boundary value problem for the multidimensional Helmholtz's equation with one singular coefficient. [ABSTRACT FROM AUTHOR]