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