Further applications of the G function integral method

TL;DR

This paper extends the G function integral method to derive new hypergeometric identities, including transformations and summations, by applying it to cubic and degenerate Miller-Paris transformations.

Contribution

It introduces novel hypergeometric transformation and summation formulas using the G function integral method, expanding its applicability to more complex functions.

Findings

New transformation formulas for hypergeometric functions

Summation formulas derived for generalized hypergeometric functions

An alternative reduction approach avoiding summation formulas

Abstract

In our recent work we proposed a generalization of the beta integral method for derivation of the hypergeometric identities which can by analogy be termed "the G function integral method". In this paper we apply this technique to the cubic and the degenerate Miller-Paris transformations to get several new transformation and summation formulas for the generalized hypergeometric functions at a fixed argument. We further present an alternative approach for reducing the right hand sides resulting from our method to a single hypergeometric function which does not require the use of summation formulas.