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

TL;DR

This paper evaluates a specific alternating multiple $t$ value using hypergeometric functions and relates it to well-known constants like $ ext{log}(2)$, $ ext{zeta}$, and $eta$, with implications for motivic mathematics.

Contribution

It provides a new explicit evaluation of a class of alternating multiple $t$ values in terms of classical constants, connecting hypergeometric asymptotics with multiple zeta values.

Findings

Explicit evaluation of $ t^{ ext{*,V}}( ext{ extbackslash overline{1}}^a, 1, ext{ extbackslash overline{1}}^b) $ in terms of $ ext{log}(2)$, $ ext{zeta}$, and $eta$.

Use of hypergeometric function asymptotics to derive the evaluation.

Discussion of potential motivic applications and conjectures.

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.