Arithmetic Constraints on Hypergeometric Identities

TL;DR

This paper investigates the arithmetic constraints that govern the existence of hypergeometric identities involving gamma functions, revealing why such identities are rare and highlighting the conditions under which they occur.

Contribution

It introduces a framework to understand the arithmetic restrictions on hypergeometric identities, clarifying the sporadic nature of their occurrence in special functions literature.

Findings

Identifies specific arithmetic conditions necessary for hypergeometric identities.

Demonstrates the constraints within a particular data region.

Explains the rarity of such identities due to these constraints.

Abstract

The standard literature on special functions contains a lot of hypergeometric identities involving products and quotients of gamma functions, but still the occurrence of such identities is a sporadic phenomenon. This is because the existence of them is constrained by severe arithmetic conditions. We demonstrate this kind of constraints by focusing on a certain data region where the essential nature of the issue comes out clearly.