On q-asymptotics for linear q-difference-differential equations with Fuchsian and irregular singularities

TL;DR

This paper develops methods to construct actual solutions for certain q-difference-differential equations with singularities, using q-Borel and Laplace transforms, and analyzes their q-asymptotic behavior including small divisors effects.

Contribution

It introduces a novel approach combining q-Borel and Laplace transforms to handle small divisors in q-difference equations with singularities, establishing q-asymptotic expansions.

Findings

Constructed holomorphic solutions with formal q-asymptotics.

Identified the impact of small divisors on q-Gevrey bounds.

Demonstrated the increase in q-exponential growth due to singularities.

Abstract

We consider a Cauchy problem for some family of q-difference-differential equations with Fuchsian and irregular singularities, that admit a unique formal power series solution in two variables t and z 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 the complex plane, is the formal solution. The small divisors's 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.