Darboux evaluations of algebraic Gauss hypergeometric functions

Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality九州大学21世紀COEプログラム「機能数理学の構築と展開」This paper presents explicit expressions for algebraic Gauss hypergeometric functions. We consider solutions of hypergeometric equations with the tetrahedral, octahedral and icosahedral monodromy groups. Conceptually, we pull-back such a hypergeometric equation onto its Darboux curve so that the pull-backed equation has a cyclic monodromy group. Minimal degree of the pull-back coverings is 4, 6 or 12 (for the three monodromy groups, respectively). In explicit terms, we replace the independent variable by a rational function of degree 4, 6 or 12, and transform hypergeometric functions to radical functions