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)/π2k 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.Comment: preprin