EXPANSIONS OF KAMP ́E DE F ́ERIET HYPERGEOMETRIC FUNCTIONS

Abstract

When studying the properties of the hypergeometric functions in two variables, expansion formulas are very important, allowing one to represent a function of two variables in the form of an infinite sum of products of several hypergeometric Gaussian functions, and this in turn facilitates the process of studying the properties of functions in two variables. Burchnall and Chaundy, in 1940--41, using the symbolic method, obtained more than 15 pairs of expansions for the second-order double hypergeometric Appell and Humbert functions. In order to find expansion formulas for functions depending on three or more variables, Hasanov and Srivastava introduced symbolic operators, with the help of which they were able to expand a whole class of hypergeometric functions of several variables. Hasanov, Turaev and Choi defined so-called H-operators that make it possible to find expansions for generalized hypergeometric functions of one variable. In addition, applications of these H-operators to the expansion of the hypergeometric functions of two and three variables of second order are known. On the other hand, thanks to the Kampe de Feriet functions, solutions of the boundary value problems for some degenerate and singular partial differential equations can be written in explicit forms. In this paper, expansion formulae are obtained for the hypergeometric Kampe de Feriet functions of the superior order. Some Kampe de Feriet functions are expanded in terms of the Appell and Humbert functions as illustrative examples