On $q$-asymptotics for $q$-difference-differential equations with Fuchsian and irregular singularities
Volume 97 / 2012
Abstract
This work is devoted to the study of a Cauchy problem for a certain family of $q$-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series $\hat{X}(t,z)$ solving the problem. Under appropriate conditions, $q$-Borel and $q$-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose $q$-asymptotic expansion in $t$, uniformly for $z$ in the compact sets of $\mathbb{C}$, is $\hat{X}(t,z)$. The small divisors phenomenon owing to the Fuchsian singularity causes an increase in the order of $q$-exponential growth and the appearance of a subexponential Gevrey growth in the asymptotics.
