Darboux evaluations for hypergeometric functions with the projective monodromy PSL(2,F7)

TL;DR

This paper develops Darboux evaluations for hypergeometric functions with the projective monodromy group PSL(2,F7), simplifying their expressions and deriving modular evaluations, advancing understanding of algebraic hypergeometric functions.

Contribution

It extends Darboux evaluations to 3F2 hypergeometric functions with PSL(2,F7) monodromy, providing new radical and modular expressions.

Findings

Darboux evaluations for 3F2-functions with PSL(2,F7) monodromy derived

Radical expressions for these hypergeometric functions established

Modular evaluations of the functions obtained

Abstract

Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations were already considered for algebraic 2F1-functions. This article presents Darboux evaluations for the classical case of 3F2-functions with the projective monodromy group PSL(2,F7). As an application, appealing modular evaluations of the same 3F2-functions are derived.