  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.
