In order to push the study of solutions of nonlinear differential equations involved in quantum electrodynamics 1 , we focus here on combinatorial aspects of their renormalization at {0, 1, +∞}.
Extending Eulerian polynomials and Faulhaber's formula 1 , we study several combinatorial aspects of harmonic sums and polylogarithms at non-positive multi-indices as well as their structure. Our techniques are based on the combinatorics of noncommutative generating series in the shuffle Hopf algebras giving a global process to renormalize the divergent polyzetas at non-positive multi-indices.
Extending the Eulerian functions, we study their relationship with zeta function of several variables. In particular, starting with Weierstrass factorization theorem (and Newton-Girard identity) for the complex Gamma function, we are interested in the ratios of ζ(2k)/π 2k and their multiindexed generalization, we obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas values, by means of some combinatorics of words and noncommutative rational series. The same frameworks also allow to study the independence of a family of eulerian functions.Résumé. -(Familles de fonctions eulériennes impliquées dans la régularisation de polyzêtas divergents) En généralisant les fonctions euleriennes, nous étudions leurs relations avec la fonction zêta en plusieurs variables. En particulier, à partir du théorème de factorisation de Weierstrass (et l'identité de Newton-Girard) pour la fonction Gamma complexe, nous nous intéressons aux rapports ζ(2k)/π 2k et leurs généralisations. Nous obtenons une situation analogue et nous tirerons quelques conséquences sur une structure de l'algèbre des valeurs polyzêtas, au moyen de la combinatoire des mots et des séries rationnelles en variables non commutatifs. Le même cadre de travail permet également d'étudier l'indépendance d'une famille de fonctions euleriennes.
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