Identification of formal power–logarithmic expansions for solutions to $q$–difference equations

Abstract

We consider an algebraic $q$–difference equation. We propose a sufficient condition for the existence of a formal power–logarithmic expansion in the vicinity of zero of the solution to such an equation. We apply this sufficient condition to construct the formal expansion of a solution to a certain $q$–difference analogue of the fifth Painlev´e equation for particular values of the parameters in the equations. We consider two different values of $q$, which lead to qualitatively different formal asymptotic expansions for the solutions.