On Hypergeometrics 3F2(1) - A Review

TL;DR

This paper systematically applies known relations to hypergeometric sums 3F2(1), discovering 62 new sums, correcting errors, and providing an algorithmic approach to find contiguous elements of classical theorems.

Contribution

It introduces a systematic method to generate new hypergeometric sums and simplifies the process of finding contiguous relations, extending previous research.

Findings

62 new hypergeometric sums identified

Errors in existing literature corrected

An algorithm for finding contiguous sums is proposed

Abstract

By systematically applying ten well-known and inequivalent two-part relations between hypergeometric sums 3F2(...|1) to the published database of all such sums, 62 new sums are obtained. The existing literature is summarized, and many purportedly novel results extracted from that literature are shown to be special cases of these new sums. The general problem of finding elements contiguous to Watson's, Dixon's and Whipple's theorems is reduced to a simple algorithm suitable for machine computation. Several errors in the literature are corrected or noted. The present paper both summarizes and extends a previous work on this subject.