Symmetry results for multiple $t$-values

TL;DR

This paper proves a symmetry relation for multiple t-values, showing that for compositions of length at least 3, t(I) equals a sign times t(reverse I) modulo products, and applies this to derive explicit formulas.

Contribution

It establishes a new symmetry property for multiple t-values under reversal of compositions, extending known results and providing explicit formulas for various classes.

Findings

Symmetry relation t(I)=(-1)^{n-1} t(reverse I) mod products for compositions of length n≥3.

Explicit formulas for certain classes of multiple t-values.

Extension of symmetry results to interpolated multiple t-values.

Abstract

For a composition $I$ whose first part exceeds 1, we can define the multiple $t$-value $t(I)$ as the sum of all the terms in the series for the multiple zeta value $\zeta(I)$ whose denominators are odd. In this paper we show that if $I$ is composition of $n\ge 3$, then $t(I)=(-1)^{n-1}t(\bar I)$ mod products, where $\bar I$ is the reverse of $I$, and both sides are suitably regularized when $I$ ends in 1. This result is not true for multiple zeta values, though there is an argument-reversal result that does hold for them (and for multiple $t$-values as well). We actually prove a more general version of this result, and then use it to establish explicit formulas for several classes of multiple $t$-values and interpolated multiple $t$-values.