نبذة مختصرة : We consider analytic and formal solutions of certain family of q-difference-differential equations under the action of a complex perturbation parameter. The previous study (Lastra and Malek in Adv. Differ. Equ. 2015:344, 2015) provides information in the case where 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 (Dreyfus in Int. Math. Res. Not. 15:6562-6587, 2015, 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, which can be described by means of a Newton polygon.
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