On confluent forms of the one hypergeometric function of three variables and their applications

Abstract

Hypergeometric functions are divided into complete and confluent functions. For the first time, Srivastava and Karlsson described a method for constructing a set of all possible triple Gaussian hypergeometric series and compiled a table showing definitions and areas of convergence for 205 different complete series (Srivastava-Karlsson List) depending on three variables. Several authors subsequently obtained various integral representations and transformations for the functions proposed by Srivastava and Karlsson. In this work we compile integral representations and transformation formulas for all confluent forms of one complete hypergeometric function in three variables from the Srivastava-Karlsson List.
To prove integral representations for 8 confluent hypergeometric functions of three variables, properties of beta and gamma functions are used. The Boltz formula allows us to derive transformation formulas for the confluent functions under consideration.