Duality and Reciprocity for Hypergeometric Series with Gamma Product Formula

TL;DR

This paper advances the understanding of non-terminating hypergeometric series with gamma product formulas by employing duality and reciprocity, resolving key conjectures and extending previous results.

Contribution

It introduces the use of duality and reciprocity to extend and strengthen results on hypergeometric series with gamma product formulas, settling prior conjectures.

Findings

Extended the region where gamma product formulas apply

Proved rationality and finiteness conjectures

Strengthened the theoretical framework for hypergeometric series

Abstract

Following a previous article we continue our study on non-terminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this article we exploit the concepts of duality and reciprocity not only to extend already obtained results to a larger region but also to strengthen themselves substantially. Among other things we are able to settle the rationality and finiteness conjectures posed in the previous article.