On the evaluation of the alternating multiple $t$ value $t(\{\overline{1}{\}}^a,1,\{\overline{1}{\}}^b)$

Abstract

We prove an evaluation for the stuffle-regularised multiple $t$ value $t^{\ast ,V}(\{\overline{1}{\}}^a,1,\{\overline{1}{\}}^b)$ in terms of $\log (2)$, $\zeta (k)$ and $\beta (k)$. This arises by evaluating the corresponding generating series using the Evans-Stanton/Ramanujan asymptotics of a zero-balanced hypergeometric function ${}_3{F}_2$, and an evaluation established by Li in an alternative approach to Zagier's evaluation of $\zeta (\{2{\}}^a,3,\{2{\}}^b)$. We end with some discussion and conjectures on possible motivic applications.

10 pages