Relations of multiple zeta values and their algebraic expression

Abstract We establish a new class of relations, which we call the cyclic sum identities, among the multiple zeta values ζ(k1,…,kl)=∑n1>⋯>nl⩾11/(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〉 .

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