Hypergeometric ${}_4F_3(1)$ with integral parameter differences

TL;DR

This paper explores the hypergeometric function ${}_4F_3(1)$ with integral parameter differences, deriving reduction formulas that simplify the function to cases involving the gamma function's logarithmic derivative.

Contribution

It introduces new reduction formulas for ${}_4F_3(1)$ with integral parameter differences, connecting complex cases to simpler, well-understood functions.

Findings

Reduction formulas for ${}_4F_3(1)$ with integral parameter differences

All cases reduce to the unit parameter difference case

Expresses the function in terms of gamma function derivatives

Abstract

In this paper we continue investigation of the hypergeometric function ${}_4F_3(1)$ as the function of its seven parameters. We deduce several reduction formulas for this function under additional conditions that one of the top parameters exceeds one of the bottom parameters by a positive integer or reversely one of the bottom parameters exceeds one of the top parameters by a positive integer or both. We show that all such cases reduce to the case of the unit parameter difference. The latter case, in turn, can be expressed in terms of certain linear combination of two series involving the logarithmic derivative of the gamma function.