On the evaluations of multiple $S$ and $T$ values of the form $S(\overset{{}_{(-)}}{2}, 1, \ldots, 1, \overset{{}_{(-)}}{1})$ and $T(\overset{{}_{(-)}}{2}, 1, \ldots, 1, \overset{{}_{(-)}}{1})$

Steven Charlton

Published 2024-03-07Version 1

Xu, Yan and Zhao showed that in even weight, the multiple $T$ value $T(2, 1, \ldots, 1, \overline{1})$ is a polynomial in $\log(2)$, $\pi$, Riemann zeta values, and Dirichlet beta values. Based on low-weight examples, they conjectured that $\log(2)$ does not appear in the evaluation. We show that their conjecture is correct, and in fact follows largely from various earlier results of theirs. More precisely, we derive explicit formulae for $T(2, 1, \ldots, 1, \overline{1})$ in even weight and $S(2, 1, \ldots, 1, \overline{1})$ in odd weight via generating series calculations. We also resolve another conjecture of theirs on the evaluations of $T(\overline{2}, 1, \ldots, 1, \overline{1})$, $S(\overline{2}, 1, \ldots, 1, 1)$, and $S(\overline{2}, 1, \ldots, 1, \overline{1})$ in even weight, by way of calculations involving Goncharov's theory of iterated integrals and multiple polylogarithms.