Analytic q-difference equations

A complex number q with 0 < |q| < 1 is fixed. By an analytic q-difference equation we mean an equation which can be represented by a matrix equation Y (z) = A(z)Y (qz) where A(z) is an invertible n× n-matrix with coefficients in the field K = C({z}) of the convergent Laurent series and where Y (z) is a vector of size n. The aim of this paper is to give an overview of our present knowledge of these equations and their solutions. Definitions and statements are presented in detail. Examples illustrate the main results. For proofs we refer to the cited literature. For section 1 and part of the following sections, the reference is [P-S]. For the later sections the reference is [P-R]. The last section presents unpublished results. The theory of linear differential equations over K (see [P-S.2], especially Chapter 10) has many of the features presented in this survey. The manuscript [R-S] contains a concise overview of analytic q-difference equations and its main purpose is to develop a theory of q-summation leading to, in our terminology, a description of a universal difference Galois group. As the authors of [R-S] note, this is only partially achieved and part of their work is still conjectural. However for a certain class of q-difference equations, namely those having at most two slopes and such that the slopes are integral, they have explicit results. An important part of the extensive literature on q-difference equations can be found in the papers cited here.