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 $ {}_3F_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.