Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

Abstract

Algebraic solutions of the sixth Painlevé equation can be constructed with the help of RS-transformations of hypergeometric equations. Construction of these transformations includes specially ramified rational coverings of the Riemann sphere and the corresponding Schlesinger transformations (S-transformations). Some algebraic solutions can be constructed from rational coverings alone, without obtaining the corresponding pullbacked isomonodromy Fuchsian system, i.e., without the S part of the RS transformations. At the same time, one and the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painlevé VI solutions. In the case of high degree coverings, construction of the S parts of the RS-transformations may represent some computational difficulties. This paper presents computation of explicit RS pullback transformations and derivation of algebraic Painlevé VI solutions from them. As an example, we present a computation of all seed solutions for pullbacks of hyperbolic hypergeometric equations. Bibliography: 26 titles.