EXPANSION FORMULAS FOR DOUBLE HYPERGEOMETRIC FUNCTIONS AND ITS APPLICATION TO THE THEORY OF SINGULAR ELLIPTIC EQUATIONS
EXPANSION FORMULAS FOR DOUBLE HYPERGEOMETRIC FUNCTIONS AND ITS APPLICATION TO THE THEORY OF SINGULAR ELLIPTIC EQUATIONS
DOI:
https://doi.org/10.52754/16948645_2023_2_149Keywords:
double hypergeometric functions, Horn list, confluent hypergeometric function, expansion formula, symbolic operators of Burchnall-Chaundy typeAbstract
It is known that the Gaussian hypergeometric function of one variable has been thoroughly investigated in all respects. Therefore, when studying the properties of hypergeometric functions of many variables, expansion formulas are very important, which make it possible to represent a function of many variables in the form of an infinite sum of products of several hypergeometric Gauss functions, and this, in turn, facilitates the process of studying the properties of functions of many variables. In the literature, 34 hypergeometric functions of two variables of order 2 (Horn List) are known, and for 11 of them in 1940-1941. Burchnall and Chaundy obtained more than 15 pairs of expansions using the symbolic method. The well-known Poole formula played an important role in the studies of Burchnall and Chaundy, but this one formula was not enough for the expansion of all functions from the Horn list. Therefore, until recently, other Horn hypergeometric functions of two variables remained undecomposed. In this paper, new symbolic operators of Burchnall-Chaundy type are introduced, their properties are studied, and an expansion for 5 Horn hypergeometric functions is established. An application of the new expansion formula to the theory of constructing fundamental solutions for singular elliptic equations is shown.
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