On finding formal power-logarithmic expansions of solutions to $q$-difference equations
Nikita Gaianov, Anastasia Parusnikova
TL;DR
The paper develops a sufficient condition for the existence of formal power-logarithmic expansions of solutions to algebraic $q$-difference equations by extending Bruno's Newton-polygon framework to the $q$-difference setting. It formalizes the relation between original and shortened equations, analyzes critical numbers arising from the linearized operator, and derives a general expansion form $y=cx^r+\sum_{k\in K} \beta_k(\log_q x)x^k$ with recursive equations for the $\beta_k$. A key contribution is the reduction of the problem to difference equations for the coefficient functions, including compatibility conditions that determine the presence or absence of logarithms, as well as a degree bound for these coefficients. The method is illustrated with a $q$-difference Painlevé V analogue, showing that different choices of $q$ lead to qualitatively different asymptotic expansions, including power-log and Puiseux-type series, highlighting the impact of $q$ on formal asymptotics in $q$-difference equations.
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 Painlevé 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 Painlevé equation.