Building Meromorphic Solutions of<i>q</i>-Difference Equations Using a Borel–Laplace Summation

Abstract

After introducing q-analogs of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace transformations of convenient orders and convenient direction in order to construct a solution of the same equation that is meromorphic on |$\mathbb {C}^{*}$|⁠. We use this theorem to construct explicitly an invertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on |$\mathbb {C}^{*}$|⁠. Moreover, when the system has two slopes and is put in the Birkhoff–Guenther normal form, we show how the solutions we compute are related to the one constructed by Ramis et al. [27].