On symmetric solutions of the fourth $q$-Painlev\'e equation

TL;DR

This paper demonstrates the existence and properties of symmetric solutions for the $q$-difference Painlevé IV equation, linking them to monodromy problems and classical special functions.

Contribution

It establishes the existence of symmetric solutions for the $q$-Painlevé IV equation and analyzes their symmetry properties and monodromy problems.

Findings

Existence of symmetric solutions for $q$-Painlevé IV.

Explicit symmetry properties of these solutions.

Connection to classical special functions and monodromy problems.

Abstract

The Painlev\'e 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 Painlev\'e equation. We focus on symmetric solutions of a $q$-difference equation known as $q\textrm{P}_{\textrm{IV}}$ or $q{\rm P}(A_5^{(1)})$ and provide their symmetry properties and solve the corresponding monodromy problem.