Representations of hypergeometric functions for arbitrary parameter values and their use

TL;DR

This paper extends integral representations of hypergeometric functions to arbitrary parameters, interprets them as regularizations involving Meijer's G function, and applies these to various special functions and inequalities.

Contribution

It introduces generalized integral representations for hypergeometric functions valid for all parameter values, linking them to Meijer's G function and exploring diverse applications.

Findings

Extended representations as regularizations involving Meijer's G function

Inverse factorial series expansion for Gauss type functions

New insights into zeros of Bessel and Kummer functions

Abstract

Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and show that the extended representations can be interpreted as examples of regularizations of integrals containing Meijer's $G$ function. Second, we give new applications of both, known and extended representations. These include: inverse factorial series expansion for the Gauss type function, new information about zeros of the Bessel and Kummer type functions, connection with radial positive definite functions and generalizations of Luke's inequalities for the Kummer and Gauss type functions.