On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities |
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Authors: | Alberto Lastra Stéphane Malek Javier Sanz |
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Institution: | 1. Universidad de Valladolid, Departamento de Análisis Matemático, Paseo Prado de la Magdalena s/n, Valladolid, Spain;2. Université de Lille, UFR de Mathématiques Pures et Appliquées, Cité Scientifique – Bât. M2, 59655 Villeneuve d?Ascq Cedex, France |
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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 (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 . The small divisors? 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. |
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