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 $\epsilon$ 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.