  The asymptotic behavior of the analytic solutions of a family of singularly
perturbed q-difference-differential equations in the complex domain is studied.
Different asymptotic expansions with respect to the perturbation parameter and
to the time variable are provided: one of Gevrey nature, and another of mixed
type Gevrey and q-Gevrey. This asymptotic phenomena is observed due to the
modification of the norm established on the space of coefficients of the formal
solution. The techniques used are based on the adequate path deformation of the
difference of two analytic solutions, and the application of several versions
of Ramis-Sibuya theorem
