Galois groups of the basic hypergeometric equations 1 by Julien Roques 20 th of August 2007

In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < |q| < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long been and is still an integral part of the mathematical literature. In particular, the Galois theory of (generalized) hypergeometric equations attracted the attention of many authors. For this issue, we refer the reader to [2, 3, 13] and to the references therein. We also single out the papers [8, 14], devoted to the calculation of some Galois groups by means of a density theorem (Ramis theorem). In this paper we focus our attention on the Galois theory of the basic hypergeometric equations, the later being natural q-analogues of the hypergeometric equations. The basic hypergeometric series φ(z) = 2φ1 (a, b; c; z) with parameters (a, b, c) ∈ (C∗)3 defined by : 2φ1 (a, b; c; z) = +∞ ∑ n=0 (a, b; q)n (c, q; q)n z