On symmetric solutions of the fourth q-Painlevé equation

Abstract

The Painlevé equations possess transcendental solutions y(t) with special initial values that are symmetric under rotation or reflection in the complex t-plane. They correspond to monodromy problems that are explicitly solvable in terms of classical special functions. In this paper, we show the existence of such solutions for a q-difference Painlevé equation. We focus on symmetric solutions of a q-difference equation known as qPIV or qP(A5 (1)) and provide their symmetry properties and solve the corresponding monodromy problem.