Families of eulerian functions involved in regularization of divergent polyzetas

TL;DR

This paper explores the extension of Eulerian functions and their connection to multivariable zeta functions, revealing new algebraic structures and independence properties through combinatorial methods.

Contribution

It introduces a novel framework linking Eulerian functions to multivariable zeta functions and analyzes their algebraic and independence properties using combinatorics.

Findings

Established relationships between Eulerian functions and multivariable zeta functions.

Identified algebraic structures of polyzeta values.

Analyzed independence of Eulerian function families.

Abstract

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 $\zeta(2k)/\pi^{2k}$ and their multiindexed generalization, we will obtain an analogue situation and draw some consequences about a structure of the algebra of polyzetas values, by means of some combinatorics of noncommutative rational series. The same combinatorial frameworks also allow to study the independence of a family of eulerian functions.