Analytic solutions of $q$-$P(A_1)$ near its critical points

TL;DR

This paper derives analytic solutions near critical points for a specific $q$-discrete Painlevé equation, connecting these solutions to classical Painlevé equations in the continuum limit.

Contribution

It provides explicit expansion solutions for a $q$-discrete Painlevé equation with 7 parameters, linking discrete and classical Painlevé equations.

Findings

Solutions approach series expansions of classical sixth Painlevé in the continuum limit

Describes solutions near critical points at origin and infinity

Connects $q$-discrete and classical Painlevé equations

Abstract

For transcendental functions that solve non-linear $q$-difference equations, the best descriptions available are the ones obtained by expansion near critical points at the origin and infinity. We describe such solutions of a $q$-discrete Painlev\'e equation, with 7 parameters whose initial value space is a rational surface of type $A_1^{(1)}$. The resultant expansions are shown to approach series expansions of the classical sixth Painlev\'e equation in the continuum limit.