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Julien Roques
Classification rationnelle et confluence des systèmes aux différences singuliers réguliers
(Rational classification and confluence of regular singular difference systems)
Annales de l'institut Fourier, 56 no. 6 (2006), p. 1663-1699, doi: 10.5802/aif.2224
Article PDF | Reviews MR 2282672 | Zbl 1125.39019
Class. Math.: 39A10, 34M35, 34M40, 65Q05
Keywords: difference equations, connection matrix, differential equations, monodromy

Résumé - Abstract

By using meromorphic “characters” and “logarithms” built up from Euler’s Gamma function, and by using convergent factorial series, we will give, in a first part, a “normal form” to the solutions of a regular singular difference system. It will enable us to define a connection matrix for a regular singular system. Following one of Birkhoff’s idea, we will then study its link with the problem of rational classification of systems. In a second part, we will be interested in the confluence of fuchsian difference systems to differential systems. We will show more particularly how we can get, under some natural hypotheses, the local monodromies of a limit differential system from the connection matrices of the deformation that we consider. The use of factorial series (which can diverge as power series) distinguish regular singular difference systems from their differential and $q$-difference analogues and make their study more difficult.

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