Article

Derivation and double shuffle relations for multiple zeta values

Collège de France, Lutetia Parisorum, Île-de-France, France
Compositio Mathematica (Impact Factor: 1.04). 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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    • "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. "
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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]. "
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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.
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    • "is equivalent to Z x n in the present paper. The proof of (2.13) in [10] is summarized in the proof of [15, Lemma 3.1] (see [15, (3.11)]). "
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    ABSTRACT: In the present paper, we prove an identity for the generating function of the quadruple zeta values. Taking homogeneous parts on both sides of the identity and substituting appropriate values for the variables, we obtain the sum formula for quadruple zeta values. We also obtain its weighted analogues, which include the formulas for this case proved by Guo and Xie (2009, J. Number Theory 129, 2747--2765) and by Ong, Eie, and Liaw (2013, Int. J. Number Theory 9, 1185--1198).
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