Algebraicity of hypergeometric functions with arbitrary parameters

TL;DR

This paper classifies when generalized hypergeometric functions are algebraic, extending previous criteria to include arbitrary complex parameters with integral differences, and demonstrates the criteria on various examples.

Contribution

It generalizes the algebraicity classification of hypergeometric functions to include arbitrary complex parameters with integral differences, expanding prior interlacing criteria.

Findings

Complete classification of algebraic hypergeometric functions with arbitrary parameters

Extension of Christol and Beukers-Heckman criteria to broader parameter sets

Application of the criteria to diverse examples

Abstract

We provide a complete classification of the algebraicity of (generalized) hypergeometric functions with no restriction on the set of their parameters. Our characterization relies on the interlacing criteria of Christol (1987) and Beukers-Heckman (1989) for globally bounded and algebraic hypergeometric functions, however in a more general setting which allows arbitrary complex parameters with possibly integral differences. We also showcase the adapted criterion on a variety of different examples.