On finding formal power-logarithmic expansions of solutions to $q$-difference equations

TL;DR

This paper establishes a sufficient condition for formal power-logarithmic expansions of solutions to algebraic q-difference equations near zero, demonstrated through an example related to the q-analogue of the fifth Painlevé equation.

Contribution

It introduces a new criterion for the existence of formal expansions in q-difference equations and applies it to a specific Painlevé analogue for different q-values.

Findings

Different q-values lead to qualitatively different formal asymptotic expansions.

The sufficient condition effectively constructs formal solutions near zero.

Application to Painlevé equations demonstrates practical utility.

Abstract

An algebraic $q$-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this sufficient condition for constructing a formal expansion of a solution to a certain $q$-difference analogue of the fifth Painlev\'{e} equation for specific values of the equation parameters is given; two different values of the number $q$ are considered, leading to qualitatively different formal asymptotic expansions of the solutions of the fifth Painlev\'{e} equation.