A. Lastra, S. Malek, J. Sanz
On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities
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ABSTRACT.  We consider a Cauchy problem for some family of linear q-difference differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables for given formal power series initial conditions. Under suitable conditions and by the application of certain q-Borel and Laplace transforms, 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 is the latter formal series. 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.\medskip