Generalisations of multiple zeta values to rooted forests

TL;DR

This paper introduces a new generalization of multiple zeta values to rooted forests, providing series representations, algebraic properties, and explicit formulas, connecting them to conical zeta values and existing results.

Contribution

It develops a series representation for shuffle arborified zeta values, establishing their algebraic structure and relation to conical zeta values, and extends known results to Mordell-Tornheim zeta values.

Findings

Series representation for convergent arborified zeta values

Connection between arborified and conical zeta values

Explicit formulas and algebraic properties derived

Abstract

We show that any convergent (shuffle) arborified zeta value admits a series representation. This justifies the introduction of a new generalisation to rooted forests of multiple zeta values, and we study its algebraic properties. As a consequence of the series representation, we derive elementary proofs of some results of Bradley and Zhou for Mordell-Tornheim zeta values and give explicit formulas. The series representation for shuffle arborified zeta values also implies that they are conical zeta values. We characterise which conical zeta values are arborified zeta values and evaluate them as sums of multiple zeta values with rational coefficients.