Computing differential Galois groups of second-order linear $q$-difference equations

TL;DR

This paper uses differential Galois theory to compute the Galois groups of second-order linear q-difference equations, revealing relations among solutions and applying to knot invariants like the colored Jones polynomial.

Contribution

It extends differential Galois theory to second-order q-difference equations with rational coefficients, enabling explicit computation of their Galois groups.

Findings

Computed Galois groups for specific q-difference equations

Identified polynomial differential relations among solutions

Applied methods to knot theory invariants like the colored Jones polynomial

Abstract

We apply the differential Galois theory for difference equations developed by Hardouin and Singer to compute the differential Galois group for a second-order linear $q$-difference equation with rational function coefficients. This Galois group encodes the possible polynomial differential relations among the solutions of the equation. We apply our results to compute the differential Galois groups of several concrete $q$-difference equations, including for the colored Jones polynomial of a certain knot.