ar X iv : m at h / 00 10 14 0 v 1 [ m at h . Q A ] 1 3 O ct 2 00 0 RELATIONS OF MULTIPLE ZETA VALUES AND THEIR ALGEBRAIC EXPRESSION
نویسندگان
چکیده
We establish a new class of relations among the multiple zeta values ζ(k1, . . . , kl) = ∑ n1>···>nl≥1 1 n k1 1 · · ·n kl k , which we call the cyclic sum identities. 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 the noncommutative polynomial ring Q〈x, y〉, and also using an action of the Hopf algebra of quasi-symmetric functions on Q〈x, y〉. 1. Multiple zeta values. Let k1, k2, . . . , kl be positive integers with k1 > 1. The multiple zeta value ζ(k1, k2 . . . , kl) (of weight k1 + · · ·+ kl and length l) associated with this sequence is the sum of the convergent l-fold infinite series
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