On parametric $0$-Gevrey asymptotic expansions in two levels for some linear partial $q$-difference-differential equations

TL;DR

This paper develops a new asymptotic representation for solutions to certain singularly perturbed q-difference-differential equations, revealing two levels linked to coefficient domain decay, and introduces a multilevel Ramis-Sibuya theorem.

Contribution

It introduces a novel two-level asymptotic expansion framework and a multilevel Ramis-Sibuya theorem for complex q-difference-differential equations.

Findings

New asymptotic representation with two levels

Extension of Ramis-Sibuya theorem to multilevel setting

Application to singularly perturbed q-difference equations

Abstract

A novel asymptotic representation of the analytic solutions to a family of singularly perturbed $q-$difference-differential equations in the complex domain is obtained. Such asymptotic relation shows two different levels associated to the vanishing rate of the domains of the coefficients in the formal asymptotic expansion. On the way, a novel version of a multilevel sequential Ramis-Sibuya type theorem is achieved.