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