Finite multiple zeta values associated with 2-colored rooted trees

TL;DR

This paper introduces finite multiple zeta values linked to 2-colored rooted trees, demonstrating they can be expressed as linear combinations of standard FMZVs and providing a new proof of their shuffle relations.

Contribution

It generalizes Kamano's work by defining FMZVs for 2-colored rooted trees and proves their expressibility as linear combinations of usual FMZVs.

Findings

FMZVs associated with 2-colored rooted trees can be explicitly expressed as integer linear combinations of standard FMZVs.

The paper provides a new proof of the shuffle relation for FMZVs.

The results extend the understanding of FMZVs and their algebraic relations.

Abstract

We define finite multiple zeta values (FMZVs) associated with some combinatorial objects, which we call 2-colored rooted trees, and prove that FMZVs associated with 2-colored rooted trees satisfying certain mild assumptions can be written explicitly as $\mathbb{Z}$-linear combinations of the usual FMZVs. Our result can be regarded as a generalization of Kamano's recent work on finite Mordell-Tornheim multiple zeta values. As an application, we will give a new proof of the shuffle relation of FMZVs, which was first proved by Kaneko and Zagier.