Thomas Dreyfus, Alberto Lastra, Stephane Malek
On the multiple-scale analysis for some linear partial q-difference and differential equations with holomorphic coefficients
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ABSTRACT.  The analytic and formal solutions of certain family of q-difference-differential equations under the action of a complex perturbation parameter is considered. In a previous study the authors have provided 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 recent paper of the first author, where he 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.