DE L'ALG EBRE DES DE RIEMANN MULTIVARI EES A L'ALG EBRE DES DE HURWITZ MULTIVARI EES

The theory of noncommutative rational power series allows to ex- press as iterated integrals some generating series associated to polylogarithms and polyz^ etas, also called MZV's (multiple zeta values : a generalization of the Riemann function). We introduce the Hurwitz polyz^ etas, as a multivalued generalization of the classical Hurwitz function. They are in fact generating series of the classical polyz^ etas in commuting variables. Based on the shue product of noncommuta- tive rational series, explicit formulae are given for computing the product of these generating series. We dene also another shue product for the Hurwitz polyz^ etas. This structure allows us to produce a new algorithm for computing the coloured polyz^ etas relations, by mean of Dirichlet generating series associated to the periodic sequences of numbers. Concerning the regularization of divergent polyz^ etas, we give explicit syntaxic formulae based on the combinatorics of words. As application we compute the Arakawa-Kaneko integrals in terms of polyz^ etas. R esum e. La th eorie des s eries rationnelles en variables non commutatives permet