Mathematische Zeitschrift

, Volume 268, Issue 3–4, pp 993–1011

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

  • Yasushi Komori
  • Kohji Matsumoto
  • Hirofumi Tsumura


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.


Multiple zeta values Witten zeta-functions Root systems 

Mathematics Subject Classification (2000)

11M41 17B20 40B05 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Yasushi Komori
    • 1
  • Kohji Matsumoto
    • 2
  • Hirofumi Tsumura
    • 3
  1. 1.Department of MathematicsRikkyo UniversityTokyoJapan
  2. 2.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  3. 3.Department of Mathematics and Information SciencesTokyo Metropolitan UniversityHachioji, TokyoJapan

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