(Strange) gamma evaluations

TL;DR

This paper surveys algorithmic strategies for obtaining closed-form gamma evaluations of hypergeometric functions, focusing on Ebisu's contiguous-relations approach and Zeilberger's telescoping framework. It highlights how both methods reduce parameter shifts to relations among a base hypergeometric function and its derivative (or neighboring shifts), with admissible choices leading to gamma-product evaluations. The discussion extends to WZ seeds, higher-rank reductions, and $q$-extensions, showing how these ideas yield numerous known and new identities while exposing limits in scope and a lack of a universal proof framework for certain rigid identities. The work emphasizes the interplay between contiguous relations, WZ theory, and Clausen-type constructions, and points to open questions about generalizability and systematic discovery of gamma evaluations.

Abstract

We review "creative" strategies of closed-form evaluations of hypergeometric functions.