Galois groups of the Lie-irreducible generalized $q$-hypergeometric equations of order three with $q$-real parameters : an approach using a density theorem

TL;DR

This paper computes the difference Galois groups of specific third-order q-hypergeometric equations with real parameters, revealing they always contain SL(3,C), using a density theorem approach.

Contribution

It introduces a method to determine Galois groups of q-hypergeometric equations of order three with real parameters, highlighting their automatic inclusion of SL(3,C).

Findings

Galois groups contain SL(3,C) for these equations.

The approach uses Sauloy's density theorem.

Results contrast with the differential case.

Abstract

In this paper we compute the difference Galois groups of the Lie-irreducible regular singular generalized q-hypergeometric equations of order 3 with q-real parameters by using a density theorem due to Sauloy. In contrast with the differential case, we show that these groups automatically contain the special linear group SL(3,C).