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

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.

arXiv comment40 pages

References in corpus (9)

• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• Differential Tannakian Categories
• Unipotent differential algebraic groups as parameterized differential Galois groups
• Galois groups for integrable and projectively integrable linear difference equations
• Splitting fields and general differential Galois theory
• Desingularization in the $q$-Weyl algebra
• Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation
• Computing the differential Galois group of a parameterized second-order linear differential equation
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

Cited by in corpus (1)

• Mahler discrete residues and summability for rational functions