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 X̂(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 ℂ, is X̂(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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc97-0-5,
author = {Alberto Lastra and St\'ephane Malek and Javier Sanz},
title = {On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities},
journal = {Banach Center Publications},
volume = {97},
year = {2012},
pages = {73-90},
zbl = {06124781},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc97-0-5}
}
Alberto Lastra; Stéphane Malek; Javier Sanz. On q-asymptotics for q-difference-differential equations with Fuchsian and irregular singularities. Banach Center Publications, Tome 97 (2012) pp. 73-90. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc97-0-5/