Symmetry results for 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.

36 pages

References in corpus (2)

• Quasi-shuffle products revisited
• The parity theorem for multiple polylogarithms