Local analytic classification of q-difference equations

TL;DR

This paper advances the classification of q-difference equations by developing new descriptions of their moduli space, extending normal forms, and introducing a novel summation theory, thereby completing Birkhoff's program.

Contribution

It provides three distinct descriptions of the moduli space of isoformal analytic classes for q-difference equations, including extended normal forms and a new summation theory.

Findings

Extended Birkhoff-Guenter normal forms

q-analogues of Birkhoff-Malgrange-Sibuya theorems

A new theory of summation for q-difference equations

Abstract

We essentially achieve Birkhoff's program for q-difference equations by giving three different descriptions of the moduli space of isoformal analytic classes. This involves an extension of Birkhoff-Guenter normal forms, q-analogues of the so-called Birkhoff-Malgrange-Sibuya theorems and a new theory of summation. The results were announced in [C. R. Math. Acad. Sci.] and in various seminars and conferences between 2004 and 2006.