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Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems

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Abstract

The shuffle product plays an important role in the study of multiple zeta values (MZVs). This is expressed in terms of multiple integrals, and also as a product in a certain non-commutative polynomial algebra over the rationals in two indeterminates. In this paper, we give a new interpretation of the shuffle product. In fact, we prove that the procedure of shuffle products essentially coincides with that of partial fraction decompositions of MZVs of root systems. As an application, we give a proof of extended double shuffle relations without using Drinfel’d integral expressions for MZVs. Furthermore, our argument enables us to give some functional relations which include double shuffle relations.

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Authors and Affiliations

  1. Department of Mathematics, Rikkyo University, Nishi-Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan

    Yasushi Komori

  2. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

    Kohji Matsumoto

  3. Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Hirofumi Tsumura

Authors
  1. Yasushi Komori
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  2. Kohji Matsumoto
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  3. Hirofumi Tsumura
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Correspondence to Hirofumi Tsumura.

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Komori, Y., Matsumoto, K. & Tsumura, H. Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems. Math. Z. 268, 993–1011 (2011). https://doi.org/10.1007/s00209-010-0705-6

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  • DOI: https://doi.org/10.1007/s00209-010-0705-6

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