%0 Journal Article %A Lastra Sedano, Alberto %A Sanz, Javier %A Malek, Stephane %T On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities %D 2012 %@ 0022-0396 %U http://hdl.handle.net/10017/41470 %X 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 X (t, 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 , is X (t, z) . The small divisorsʼ 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. %K q-Difference-differential equations %K q-Laplace transform %K Formal power series solutions %K q-Gevrey asymptotic expansions %K Small divisors %K Fuchsian and irregular singularities %K Matemáticas %K Mathematics %~ Biblioteca Universidad de Alcala