Stephane Malek
On parametric Gevrey asymptotics for a q-analog of some linear initial value problem
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ABSTRACT. We study an inhomogeneous linear q-difference differential Cauchy problem, with a complex perturbation parameter. This problem is seen as a q-analog of an initial value problem recently investigated by the author and A. Lastra. Here a comparable
result is achieved, namely we construct a finite set of holomorphic solutions on a common bounded open sector in time at the origin, on a given strip in space, when the parameter belongs to a well selected set of open bounded sectors whose union covers a neighborhood of 0 in the complex domain. These solutions are constructed through a continuous version of a q-Laplace transform of some order k and Fourier inverse map of some function with q-exponential growth of order k on adequate unbounded sectors and with exponential decay in the Fourier
variable. Moreover, by means of a q-analog of the classical Ramis-Sibuya theorem, we prove that they share a common formal power series
(that generally diverge) in the perturbation parameter as q-Gevrey asymptotic expansion of order 1/k.