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Local analytic classification of $$q$$-difference equations. (English) Zbl 1320.39008
Astérisque 355. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-775-9/pbk). vi, 151 p. (2013).
This monograph is concerned with a large problem begun by G. D. Birkhoff in the beginning of the XXth century: the generalized Riemann problem for linear differential equations and the allied problems for linear difference and $$q$$-difference equations. Such problems are called Riemann-Hilbert-Birkhoff problems.
The authors classify analytically isoformal $$q$$-difference equations. They consider analytic $$q$$-difference modules together with an isomorphism of their formalization with a given formal $$q$$-difference module. They also consider equivalences preserving this additional structure.
The first chapter is a brief presentation of the book and of the notations.
In the second one, the general settings of the problem are established and two important tools are introduced: the Newton polygon and the slope filtration.
The first classification problem for $$q$$-difference equations arises in Chapter 3. From now on, one supposes that the slopes of the Newton polygon are integers.
Chapter 4 gives some extensions to $$q$$-difference equations of the so-called Birkhoff-Malgrange-Sibuya theorems. Such a result is applied to the classification problem for $$q$$-difference equations.
In Chapter 5, one develops a summation process for $$q$$-Gevrey divergent series. The $$q$$-Gevrey analogue of the classical theorems is stated. The theory is applied to $$q$$-difference equations and to their classification. The summability of solutions is proved. As a consequence, one obtains the existence of asymptotic solutions and the description of the Stokes phenomenon.
The sixth chapter gives some additional information on the geometry of the space of analytic isoformal classes through its identification with a certain cohomology set.
Chapter 7 deals with some examples in relation with $$q$$-special functions, either linked to modular functions or to confluent basic hypergeometric series.
The book ends with an appendix devoted to the classification of all finitely filtered objects with fixed associated graded object, up to equivalence compatible with the graduation.

##### MSC:
 39A13 Difference equations, scaling ($$q$$-differences) 39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 40B99 Multiple sequences and series 34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain