A note on a hypergeometric transformation formula due to Slater with an application

TL;DR

This paper revisits a hypergeometric transformation formula by Slater, providing a corrected statement, an alternative proof, and deriving a new transformation for specific hypergeometric series involving ${}_5F_4$ and ${}_3F_2$ functions.

Contribution

It offers a corrected version and an alternative proof of Slater's hypergeometric transformation formula, and introduces a new transformation relating ${}_5F_4$ and ${}_3F_2$ series.

Findings

Corrected and clarified hypergeometric transformation formula.

Derived a new transformation for ${}_5F_4(-1)$ series.

Expressed a ${}_5F_4$ series as a combination of two ${}_3F_2$ series.

Abstract

In this note we state (with minor corrections) and give an alternative proof of a very general hypergeometric transformation formula due to Slater. As an application, we obtain a new hypergeometric transformation formula for a ${}_5F_4(-1)$ series with one pair of parameters differing by unity expressed as a linear combination of two ${}_3F_2(1)$ series.