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We consider a Cauchy problem for some family of q-difference-differentialequations with Fuchsian and irregular singularities, that admit a unique formalpower series solution in two variables t and z for given formal power seriesinitial conditions. Under suitable conditions and by the application of certainq-Borel and Laplace transforms (introduced by J.-P. Ramis and C. Zhang), we areable to deal with the small divisors phenomenon caused by the Fuchsiansingularity, and to construct actual holomorphic solutions of the Cauchyproblem whose q-asymptotic expansion in t, uniformly for z in the compact setsof the complex plane, is the formal solution. The small divisors's effect is anincrease in the order of q-exponential growth and the appearance of a power ofthe factorial in the corresponding q-Gevrey bounds in the asymptotics.

On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities