Classification rationnelle et confluence des systemes aux differences singuliers reguliers

Details Classification rationnelle et confluence des systemes aux differences singuliers reguliers Classification rationnelle et confluence des systemes aux differences singuliers reguliers

2006-10-19

arxiv.org/abs/math/0610596v1

Mathematics

Classical Analysis and ODEs

Scientific paper

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 pat, a "normal form" to the solutions of a singular regular system. It will enable us to define a connexion 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 system. 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. En choisissant des "caracteres" et des "logarithmes", meromorphes sur le plan complexe, construits a l'aide de la fonction Gamma d'Euler, et en utilisant des series de factorielles convergentes, nous sommes en mesure, dans une premiere partie, de donner une "forme normale" pour les solutions d'un systeme aux differences singulier regulier. Nous pouvons alors definir une matrice de connexion d'un tel systeme. Nous etudions ensuite, suivant une idee de G.D. Birkhoff, le lien de celles-ci avec le probleme de la classification rationnelle des systemes. Dans une deuxieme partie, nous nous interessons la confluence des systemes aux differences fuchsiens vers les systemes differentiels. Nous montrons en particulier comment, sous certaines hypotheses naturelles, on peut reconstituer les monodromies locales d'un systeme differentiel limite a partir des matrices meromorphes de connexion des deformations considerees. Le point central, qui distingue en profondeur les systemes aux differences singuliers reguliers de leurs homonymes differentiels ou aux q-differences et qui rend leur etude plus complexe, est la necessaire utilisation de series de factorielles (qui peuvent diverger en tant que series de puissances).

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