Reductions of particular hypergeometric functions $_3F_2(a,a+1/3,a+2/3;p/3,q/3;\pm 1)$

TL;DR

This paper presents new reductions of specific hypergeometric functions $_3F_2( imes)$ into elementary functions, introduces a novel method involving alternating binomial sums, and derives a functional equation without using the WZ method.

Contribution

The paper introduces a new approach to reduce hypergeometric functions $_3F_2( imes)$ and derives a functional equation for related binomial sums, avoiding the WZ summation technique.

Findings

New reductions of $_3F_2( imes)$ functions into elementary functions

A functional equation for three alternating binomial sums

A method involving solving for three sums simultaneously

Abstract

We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we simultaneously solve for three alternating binomial sums, may be new. We obtain a functional equation holding for all three of this set of alternating binomial sums. Using successive derivatives, we show how related chains of $_3F_2(\pm 1)$ values may be obtained. It may be emphasized that we make no reliance on the WZ method for hypergeometric summation. Additional material on Pochhammer symbols and certain of their products is presented in an Appendix to supplement the pedagogical content of the paper.