Dihedral Gauss hypergeometric functions

TL;DR

This paper provides explicit elementary and trigonometric expressions for dihedral Gauss hypergeometric functions, simplifying their analysis by transforming their monodromy groups to cyclic groups and expressing solutions via finite sums.

Contribution

It introduces general elementary formulas and trigonometric representations for dihedral hypergeometric functions, including degenerate cases, expanding the toolkit for their analysis.

Findings

Elementary expressions involve finite bivariate sums

Trigonometric forms are derived for these functions

Degenerate cases with special monodromy groups are analyzed

Abstract

Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell's F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are considered.