Abstract
A family of linear singularly perturbed difference differential equations is examined. These equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.
Highlights
In this work, we focus on singularly perturbed linear partial qdifference differential equations which couple two categories of operators acting both on the time variable, so-called qdifference operators of irregular type and Fuchsian differential operators
The Ramis—Sibuya theorem is known as a cohomological criterion which ensures the existence of a common Gevrey asymptotic expansion of a given order for families of sectorial holomorphic functions
Proof. e proof mimics the arguments of Lemma XI-2-6 from [25] with fitting adjustment in the asymptotic expansions of the functions Ψp constructed by means of the Cauchy—Heine transform
Summary
We focus on singularly perturbed linear partial qdifference differential equations which couple two categories of operators acting both on the time variable, so-called qdifference operators of irregular type and Fuchsian differential operators. As a seminal reference concerning analytic and algebraic aspects of q-difference equations with irregular type, refer [1], and for a far reaching investigation of Fuchsian ordinary and partial differential equations, refer [2]
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