On the summability and convergence of formal solutions of linear $q$-difference-differential equations with constant coefficients

TL;DR

This paper investigates the convergence and summability of formal solutions to linear q-difference-differential equations with constant coefficients, providing characterizations based on analytic continuation and growth of initial data.

Contribution

It introduces criteria for summability and convergence of formal solutions and defines sequences that preserve summability, aiding analysis of moment differential equations.

Findings

Characterization of convergent, k-summable, and multisummable solutions

Introduction of summability-preserving sequences

Application to moment differential equations

Abstract

We consider the Cauchy problem for homogeneous linear $q$-difference-differential equations with constant coefficients. We characterise convergent, $k$-summable and multisummable formal power series solutions in terms of analytic continuation properties and growth estimates of the Cauchy data. We also introduce and characterise sequences preserving summability, which make a very useful tool, especially in the context of moment differential equations.