Article

# Derivation and double shuffle relations for multiple zeta values

Collège de France, Lutetia Parisorum, Île-de-France, France
(Impact Factor: 0.99). 03/2006; 142(02). DOI: 10.1112/S0010437X0500182X
Source: OAI

ABSTRACT

Derivation and extended double shuffle (EDS) relations for multiple zeta values (MZVs) are proved. Related algebraic structures of MZVs, as well as a version of EDS relations are also studied.

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• "We will focus on quasi-shuffle relations, directly derived from the nested sum representation (2), for example, ζ(a)ζ(b) = ζ(a, b) + ζ(b, a) + ζ(a + b). The two representations combined yield intricate relations, which are commonly refereed to as double shuffle structures underlying MZVs [10]. "
##### Article: Transfer group for renormalized multiple zeta values
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ABSTRACT: We describe in this work all solutions to the problem of renormalizing multiple zeta values at arguments of any sign in a quasi-shuffle compatible way. As a corollary we clarify the relation between different renormalizations at non-positive values appearing in the recent literature.
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• "Several of the papers mentioned above also contain formulas of a similar flavor for the values of more general " multiple zeta values " defined in terms of several integer-valued parameters; see also [14], [20], [29], [45] for related results. The results described above may not make evident why it is natural to consider ω(s) as a true Dirichlet series (that is, as a function of a complex variable s), but plenty of precedents from the history of analytic number theory suggest that this is worth doing. "
##### Article: On the number of n-dimensional representations of SU(3), the Bernoulli numbers, and the Witten zeta function
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ABSTRACT: We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number of n-dimensional representations of SU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.
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• "), where ζ X stands for the 'shuffle regularized' value, which is the constant term of the shuffle regularized polynomial defined in [19]. "
##### Article: Poly-Bernoulli numbers and related zeta functions
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ABSTRACT: We construct and study a certain zeta function which interpolates multi-poly-Bernoulli numbers at non-positive integers and whose values at positive integers are linear combinations of multiple zeta values. This function can be regarded as the one to be paired up with the $\xi$-function defined by Arakawa and the first-named author. We show that both are closely related to the multiple zeta functions. Further we define multi-indexed poly-Bernoulli numbers, and generalize the duality formulas for poly-Bernoulli numbers by introducing more general zeta functions.