Isomonodromic deformation of q-difference equations and confluence

TL;DR

This paper investigates how Fuchsian linear q-difference systems deform without changing their monodromy, examining the behavior of the Birkhoff connection matrix as q approaches 1, with applications to the q-analogue of the sixth Painlevé equation.

Contribution

It provides new insights into the isomonodromic deformation of q-difference systems and analyzes the convergence of the Birkhoff connection matrix in the context of Painlevé equations.

Findings

Behavior of Birkhoff connection matrix as q approaches 1

Convergence results for q-analogue of Painlevé VI

Characterization of isomonodromic deformations in q-difference systems

Abstract

We study isomonodromic deformation of Fuchsian linear q-difference systems. Furthermore, we are looking of the behaviour of the Birkhoff connexion matrix when q goes to $1$. We use our results to study the convergence of the Birkhoff connexion matrix that appears in the definition of the q-analogue of the sixth Painlev\'e equation.