TY - JOUR

T1 - Relations of multiple zeta values and their algebraic expression

AU - Hoffman, Michael E.

AU - Ohno, Yasuo

N1 - Funding Information:
The first author conjectured the cyclic sum theorem in August 1999, and thanks Michael Bigotte for checking it by computer against tables of known relations [1] through weight 12. The second author proved the conjecture during his stay at the Max-Planck-Institut für Mathematik in Bonn in early 2000, and he thanks Masanobu Kaneko and Don Zagier for useful discussions and the Institut for its hospitality. The second author is supported in part by the Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists.

PY - 2003/4/15

Y1 - 2003/4/15

N2 - 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).

AB - 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).

KW - Cyclic derivation

KW - Multiple zeta values

KW - Quasi-symmetric functions

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U2 - 10.1016/S0021-8693(03)00016-4

DO - 10.1016/S0021-8693(03)00016-4

M3 - Article

AN - SCOPUS:0038029820

VL - 262

SP - 332

EP - 347

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -