Five Parameter Hypergeometric 3F2(1) when One or more Parameters are Integers or Separated by Integers: Derivations, Review, Exotics and More

TL;DR

This paper reviews and simplifies many identities related to the hypergeometric function 3F2(1), especially when parameters are integers or separated by integers, and introduces new insights and techniques for evaluating these functions.

Contribution

It provides a simplified derivation of existing identities and offers a collection of general identities and techniques for evaluating 3F2(1) with specific parameter conditions, including some new results.

Findings

Simplified derivations of known identities

A collection of general evaluation techniques for 3F2(1)

Potential new results on hypergeometric evaluations

Abstract

This work was intended to be all about, and only about, hypergeometric 3F2(1). The initial goal was to revisit many identities from the literature that have been derived over the years and show that they can be obtained in a simpler way armed, with only a minimum of elementary identities. That goal has been achieved as a (patient) reader will discover. In another sense, this work is a partial review of the last half-century's worth of progress in the evaluation of a particular set of 3F2(1), in particular those cases where at least one parameter is an integer or two or more parameters are separated by an integer. The result is a collection of very general identities (or techniques) that an analyst seeking to evaluate a particular 3F2(1), might want to consider as a starting point. That is the secondary goal. Along the way however, the temptation arose to investigate at least one of the unanswered questions that others have raised. This led to a few digressions, and some possibly new results. The reader is invited to follow where curiosity led me to depart from a straightforward review of the state-of-the-art.