Computation of $RS$-pullback transformations for algebraic Painlevé VI solutions

R. Vidunas   ; A. V. Kitaev

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 23, Tome 433 (2015), pp. 131-155

Résumé

Algebraic solutions of the sixth Painlevé equation can be constructed with the help of $RS$-transformations of the hypergeometric equations. Construction of these transformations includes specially ramified rational coverings of the Riemann sphere and corresponding Schlesinger transformations ($S$-transformations). Some algebraic solutions can be constructed from rational coverings alone, without obtaining the corresponding pullbacked isomonodromy \break Fuchsian system, i.e., without $S$ 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 case of high degree coverings construction of $S$ parts of the $RS$-transformations may represent some computational difficulties. This paper presents computations of explicit $RS$-pullback transformations, and derivation of algebraic Painlevé VI solutions from them. As an example, we present computation of all seed solutions for pull-backs of hyperbolic hypergeometric equations.

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