Identities involving cyclic and symmetric sums of regularized multiple zeta values

Tomoya Machide

Vol. 286 (2017), No. 2, 307–359

DOI: 10.2140/pjm.2017.286.307

Abstract

There are two types of regularized multiple zeta values: harmonic and shuffle types. The first purpose of the present paper is to give identities involving cyclic sums of regularized multiple zeta values of both types for depth less than $5$. Michael Hoffman, in “Quasi-symmetric functions and mod $p$ multiple harmonic sums” (Kyushu Journal of Mathematics 69 (2015), 345–366) proved an identity involving symmetric sums of regularized multiple zeta values of harmonic type for arbitrary depth. The second purpose is to prove Hoffman’s identity for shuffle type. We also give a connection between the identities involving cyclic sums and symmetric sums, for depth less than $5$.

Keywords

multiple zeta value, cyclic sum, symmetric sum, group ring of symmetric group

Mathematical Subject Classification 2010

Primary: 11M32

Secondary: 16S34, 20C05

Milestones

Received: 15 December 2014

Revised: 17 December 2015

Accepted: 14 January 2016

Published: 15 January 2017