$q$-Borel-Laplace summation for $q$-difference equations with two slopes

TL;DR

This paper develops a $q$-analogue of Borel-Laplace summation to compute meromorphic gauge transformations for linear $q$-difference systems with two slopes, extending previous work to a new class of equations.

Contribution

It introduces a $q$-Borel-Laplace summation method specifically for systems with two slopes, providing a new computational approach for meromorphic gauge transformations.

Findings

Provides explicit $q$-Borel-Laplace summation formulas

Enables computation of gauge transformations in systems with two slopes

Extends previous methods to a broader class of $q$-difference equations

Abstract

In this paper, we consider linear $q$-difference systems with coefficients that are germs of meromorphic functions, with Newton polygon that has two slopes. Then, we explain how to compute similar meromorphic gauge transformations than those appearing in the work of Bugeaud, using a $q$-analogue of the Borel-Laplace summation.