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From the algebras of Riemann polyzetas to the algebra of Hurwitz polyzetas. (De l’algèbre des $$\zeta$$ de Riemann multivariées à l’algèbre des $$\zeta$$ de Hurwitz multivariées.) (French) Zbl 1035.11039
Summary: The theory of noncommutative rational power series allows to express as iterated integrals some generating series associated to polylogarithms and polyzetas, also called MZV’s (multiple zeta values: a generalization of the Riemann $$\zeta$$ function). We introduce the Hurwitz polyzetas, as a multivalued generalization of the classical Hurwitz $$\zeta$$ function. They are in fact generating series of the classical polyzetas in commuting variables. Based on the shuffle product of noncommutative rational series, explicit formulae are given for computing the product of these generating series. We define also another shuffle product for the Hurwitz polyzetas. This structure allows us to produce a new algorithm for computing the coloured polyzetas relations, by mean of Dirichlet generating series associated to the periodic sequences of numbers. Concerning the regularization of divergent polyzetas, we give explicit syntactic formulae based on the combinatorics of words. As application we compute the Arakawa-Kaneko integrals in terms of polyzetas.

##### MSC:
 11M41 Other Dirichlet series and zeta functions 11G55 Polylogarithms and relations with $$K$$-theory 11M35 Hurwitz and Lerch zeta functions 68R15 Combinatorics on words
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