A note on $\mathcal{F}_n$-multiple zeta values

TL;DR

This paper demonstrates that certain known evaluations and relations of multiple zeta values, previously valid in specific algebraic contexts, are universally valid in the broader setting of $\mathcal{F}_n$-multiple zeta values, including new proofs of the Bowman-Bradley theorem and sum formulas.

Contribution

It establishes the uniform validity of evaluations and relations of multiple zeta values across different algebraic frameworks, extending previous results.

Findings

Proved the Bowman-Bradley type theorem for $\\mathcal{S}_2$-multiple zeta values.

Established sum formulas for $\\mathcal{S}_2$-multiple zeta values.

Showed that known relations in $\\mathcal{A}_n$ and $\\mathcal{S}_n$ are valid in $\\mathcal{F}_n$-multiple zeta values.

Abstract

For several evaluations of special values and several relations known only in $\mathcal{A}_n$-multiple zeta values or $\mathcal{S}_n$-multiple zeta values, we prove that they are uniformly valid in $\mathcal{F}_n$-multiple zeta values for both the case where $\mathcal{F}=\mathcal{A}$ and $\mathcal{F}=\mathcal{S}$. In particular, the Bowman-Bradley type theorem and sum formulas for $\mathcal{S}_2$-multiple zeta values are proved.