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arXiv · 2512.19006

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

Nikita Gaianov·Anastasia Parusnikova
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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.

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Nikita Gaianov, Anastasia Parusnikova. 2025-12-22. On finding formal power-logarithmic expansions of solutions to $q$-difference equations. https://arxiv.org/abs/2512.19006

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