Relations of multiple zeta values and their algebraic expression

TL;DR

This paper introduces cyclic sum identities among multiple zeta values, providing elementary proofs and linking them to existing sum theorems, shuffle, and harmonic product relations, with connections to cyclic derivations and quasi-symmetric functions.

Contribution

It establishes a new class of relations called cyclic sum identities for multiple zeta values, with elementary proofs and connections to algebraic structures.

Findings

Cyclic sum identities imply the sum theorem for multiple zeta values.

Relations expressed via shuffle and harmonic products on the algebra of noncommutative polynomials.

Connections made between multiple zeta value relations and cyclic derivations, quasi-symmetric functions.

Abstract

We establish a new class of relations among the multiple zeta values \zeta(k_1,k_2,...,k_n), 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 of the noncommutative polynomial ring Q<x,y>, and also using an action on Q<x,y> of the Hopf algebra of quasi-symmetric functions.