Transformations and invariants for dihedral Gauss hypergeometric functions

TL;DR

This paper provides explicit formulas for invariants and transformations of dihedral hypergeometric functions, enhancing understanding of their algebraic solutions and symmetries.

Contribution

It introduces explicit expressions for quadratic monodromy invariants and describes pull-back transformations, including Klein's transformations, for dihedral hypergeometric equations.

Findings

Explicit formulas for monodromy invariants using hypergeometric sums

Pull-back transformations including Klein's for dihedral cases

Enhanced understanding of algebraic solutions of hypergeometric equations

Abstract

Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a generalization of Clausen's formula and terminating double hypergeometric sums. Besides, pull-back transformations for the dihedral hypergeometric equations are presented, including Klein's pullback transformations for the equations with a finite (dihedral) monodromy group.