Realizations of Affine Weyl Group Symmetries on the Quantum Painlevé Equations by Fractional Calculus

Abstract

We realize affine Weyl group symmetries on the Schrodinger equations for the quantum Painleve equations, by fractional calculus. This realization enables us to construct an infinite number of hypergeometric solutions to the Schrodinger equations for the quantum Painleve equations. In other words, since the Schrodinger equations for the quantum Painleve equations are equivalent to the Knizhnik–Zamolodchikov equations, we give one method of constructing hypergeometric solutions to the Knizhnik–Zamolodchikov equations.