On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

We study a family of singularly perturbed. q-difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter epsilon. Moreover, we achieve the existence of a common formal power series in.. which represents each actual solution and establish q-Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding. q-Gevrey asymptotics. A particular Dirichlet like series is studied on the way.