Relations of multiple zeta values and their algebraic expression

Michael E. Hoffman, Yasuo Ohno

Research output: Contribution to journalArticlepeer-review

61 Citations (Scopus)


We establish a new class of relations, which we call the cyclic sum identities, among the multiple zeta values ζ (k1,..., kl) = ∑n1 >⋯>nl≥1 1/(n1k1 ⋯ nkkl). These identities have an elementary proof and imply the "sum theorem" for multiple zeta values. They also have a succinct statement in terms of "cyclic derivations" as introduced by Rota, Sagan, and Stein. In addition, we discuss the expression of other relations of multiple zeta values via the shuffle and "harmonic" products on the underlying vector space h of the noncommutative polynomial ring Q(x, y), and also using an action of the Hopf algebra of quasi-symmetric functions on Q(x, y).

Original languageEnglish
Pages (from-to)332-347
Number of pages16
JournalJournal of Algebra
Issue number2
Publication statusPublished - 2003 Apr 15
Externally publishedYes


  • Cyclic derivation
  • Multiple zeta values
  • Quasi-symmetric functions

ASJC Scopus subject areas

  • Algebra and Number Theory


Dive into the research topics of 'Relations of multiple zeta values and their algebraic expression'. Together they form a unique fingerprint.

Cite this