Bowman-Bradley type theorem for finite multiple zeta values in $\mathcal{A}_2$

TL;DR

This paper extends Bowman-Bradley type theorems to finite multiple zeta values in the algebra , showing that certain sums are rational multiples of special elements, thus bridging classical and finite multiple zeta value theories.

Contribution

It generalizes Bowman-Bradley theorems from  to  for finite multiple zeta values, providing new insights into their algebraic structure.

Findings

Finite multiple zeta sums in  are rational multiples of special elements.

The result is a step closer to the original Bowman-Bradley theorem.

Partial extension from  to  in the context of finite multiple zeta values.

Abstract

Bowman and Bradley obtained a remarkable formula among multiple zeta values. The formula states that the sum of multiple zeta values for indices which consist of the shuffle of two kinds of the strings $\{1,3,\ldots,1,3\}$ and $\{2,\ldots,2\}$ is a rational multiple of a power of $\pi^2$. Recently, Saito and Wakabayashi proved that analogous but more general sums of finite multiple zeta values in an adelic ring $\mathcal{A}_1$ vanish. In this paper, we partially lift Saito-Wakabayashi's theorem from $\mathcal{A}_1$ to $\mathcal{A}_2$. Our result states that a Bowman-Bradley type sum of finite multiple zeta values in $\mathcal{A}_2$ is a rational multiple of a special element and this is closer to the original Bowman-Bradley theorem.