# On the multiple-scale analysis for some linear partial $q$-difference and differential equations with holomorphic coefficients

## TL;DR

This paper develops a new method for analyzing solutions of non-factorizable linear $q$-difference-differential equations with holomorphic coefficients, using successive $q$-Borel-Laplace transforms and Newton polygons to distinguish asymptotic levels.

## Contribution

It introduces a novel approach for handling non-factorizable equations by applying successive $q$-Borel-Laplace transforms guided by Newton polygon analysis.

## Key findings

- Successfully distinguishes multiple $q$-Gevrey asymptotic levels.
- Provides a framework for analyzing complex $q$-difference-differential equations.
- Extends previous methods to non-factorizable cases.

## Abstract

The analytic and formal solutions of certain family of $q$-difference-differential equations under the action of a complex perturbation parameter is considered. The previous study of the last two authors provides information in the case when the main equation under study is factorizable, as a product of two equations in the so-called normal form. Each of them gives rise to a single level of $q$-Gevrey asymptotic expansion. In the present work, the main problem under study does not suffer any factorization, and a different approach is followed. More precisely, we lean on the technique developed in a paper, where the first author makes distinction among the different $q$-Gevrey asymptotic levels by successive applications of two $q$-Borel-Laplace transforms of different orders both to the same initial problem and which can be described by means of a Newton polygon.