Relations for a class of terminating ${}_4F_3(4)$ hypergeometric series

TL;DR

This paper develops new relations and symmetry properties for specific terminating hypergeometric series, extends known summation formulas, and explores their connections through series reversal and limiting cases.

Contribution

It introduces new relations for terminating ${}_4F_3(4)$ series, analyzes their symmetry group, and derives formulas linking terminating and nonterminating series, expanding the theoretical framework.

Findings

Invariance group is isomorphic to symmetric group S_3

Derived formulas for sums of nonterminating series

Connected known summation formulas as limiting cases

Abstract

We derive relations for a certain class of terminating ${}_4F_3(4)$ hypergeometric series with three free parameters. The invariance group composed of these relations is shown to be isomorphic to the symmetric group $S_3$. We further study relations for terminating ${}_3F_2(4)$ series that fall under two families. By using a series reversal, we examine the corresponding terminating ${}_4F_3(1/4)$ and ${}_3F_2(1/4)$ series relations. We additionally derive formulas for the sums of the first $n+1$ terms of several nonterminating ${}_3F_2(4)$ and ${}_3F_2(1/4)$ series. We also show how certain known summation formulas for terminating ${}_2F_1(4)$ and ${}_3F_2(4)$ series follow as limiting cases of some of our relations.