Beyond the beta integral method: transformation formulas for hypergeometric functions via Meijer's G function

TL;DR

This paper generalizes the beta integral method by using Meijer's G function to derive numerous new hypergeometric identities and summation formulas, expanding the toolkit for hypergeometric function transformations.

Contribution

It introduces a novel approach replacing the beta density with Meijer's G function to generate new hypergeometric identities and summation formulas.

Findings

Derived approximately forty hypergeometric identities

Most identities are believed to be new

Obtained several new summation formulas

Abstract

The beta integral method proved itself as a simple nonetheless powerful method of generating hypergeometric identities at a fixed argument. In this paper we propose a generalization by substituting the beta density with a particular type of Meijer's G function. By application of our method to known transformation formulas we derive about forty hypergeometric identities, majority of which are believed to be new. We further apply some of these transformations to obtain several new summation formulas.