Darboux evaluations for hypergeometric functions with the projective monodromy $$\hbox {PSL}(2,\mathbb {F}_7)$$

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 have already been considered for algebraic $${}_{2}\text{ F }_{1}$$ -functions. This article presents Darboux evaluations for the classical case of $${}_{3}\text{ F }_{2}$$ -functions with the projective monodromy group $$\hbox {PSL}(2,\mathbb {F}_7)$$ . The pullback curves are of genus 0 (in the simplest case) or of genus 1. As an application of the genus 0 evaluations, appealing modular evaluations of the same $${}_{3}\text{ F }_{2}$$ -functions are derived.