A combinatorial proof of the weighted sum formula for finite and symmetric multiple zeta(-star) values

TL;DR

This paper provides an alternative combinatorial proof for the weighted sum formula applicable to both finite and symmetric multiple zeta(-star) values, expanding the understanding of their algebraic properties.

Contribution

It introduces a new combinatorial proof technique for the weighted sum formula, applicable to finite and symmetric multiple zeta(-star) values, offering a different perspective from previous proofs.

Findings

Proof valid for finite multiple zeta(-star) values

Extension to symmetric multiple zeta(-star) values

Simplifies understanding of weighted sum relations

Abstract

Hirose, Saito, and the author established the weighted sum formula for finite multiple zeta(-star) values. In this paper, we present its alternative proof. The proof is also valid for symmetric multiple zeta(-star) values.