Ratios of the Gauss hypergeometric functions with parameters shifted by integers: more on integral representations

TL;DR

This paper derives explicit integral representations for ratios of Gauss hypergeometric functions with shifted parameters, extending previous results and providing tools for analyzing their properties and applications.

Contribution

It introduces new integral representations for hypergeometric function ratios with shifted parameters, removing previous restrictions and enabling broader applications.

Findings

Explicit integral representations derived

Extended previous results by lifting parameter restrictions

Illustrated with examples and an application to products

Abstract

We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for this ratio based on a formula for its imaginary part. This work extends our recent results by lifting certain restrictions on parameters. The new representations are illustrated with a few examples and an application to products of ratios.