On multiscale Gevrey and q-Gevrey asymptotics for some linear   q-difference differential initial value Cauchy problems
  We study the asymptotic behavior of the solutions related to a singularly perturbed q-difference-differential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and q-Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution.   The proof leans on a two level novel version of Ramis-Sibuya theorem under Gevrey and q-Gevrey orders. 
Alberto Lastra and Stephane Malek
math.AP math.CV