On a q-Analog of a Singularly Perturbed Problem of Irregular Type with Two Complex Time Variables

Abstract

The analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain are constructed and studied from an asymptotic point of view with respect to the perturbation parameter. Two types of holomorphic solutions, the so-called inner and outer solutions, are considered. Each of them holds a particular asymptotic relation with the formal ones in terms of asymptotic expansions in the perturbation parameter. The growth rate in the asymptotics leans on the - 1 -branch of Lambert W function, which turns out to be crucial.