Some Remarks on Maesaka-Seki-Watanabe's Formula for the Multiple Harmonic Sums

TL;DR

The paper investigates the Maesaka–Seki–Watanabe (MSW) formula, establishing that for any index $\boldsymbol{k}$ and $N>0$, $\zeta_{<N}(\boldsymbol{k})=\zeta_{<N}^{\flat}(\boldsymbol{k})$, and extends this discrete equality to star sums, i.e., $\zeta_{<N}^{\star}(\boldsymbol{k})=\zeta_{<N}^{\star\flat}(\boldsymbol{k})$. It then generalizes these ideas to diagonally constant Schur indices via a connected-sum approach and a determinant trick, showing $\zeta_{<N}(\boldsymbol{k})=\zeta_{<N}^{\flat}(\boldsymbol{k})$ for anti-hook diagrams and linking to the 2-poset integral framework of HMO. The work further explicates Hoffman's duality by deriving a finite-sum form $H_{<N}(\boldsymbol{k})=H_{<N}^{\flat}(\boldsymbol{k})$ using the MSW star relation, and relates these discrete MSW identities to Kawashima's interpolation framework, notably proving $G_{\overleftarrow{\boldsymbol{k}}}(N-1)=\zeta_{<N}^{\star\flat}(\boldsymbol{k})$. Collectively, these results broaden the MSW phenomenon to star and Schur variants and reveal deep connections with Hoffman's duality and Kawashima's identity, providing new finite-sum representations and alternative proofs within the theory of multiple zeta values.

Abstract

Recently, Maesaka, Seki and Watanabe discovered a surprising equality between multiple harmonic sums and certain Riemann sums which approximate the iterated integral expression of the multiple zeta values. In this paper, we describe the formula corresponding to the multiple zeta-star values and, more generally, to the Schur multiple zeta values of diagonally constant indices. We also discuss the relationship of these formulas with Hoffman's duality identity and an identity due to Kawashima.

Theorems & Definitions (25)

• Theorem 1.1: MSW
• Theorem 2.1
• proof
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Example 3.4
• Definition 3.5
• Theorem 3.6
• Remark 3.7