Integral Evaluation of Odd Euler Sums, Multiple $t$-Value $t\left(3,2,\ldots,2\right)$ and Multiple Zeta Value $\zeta(3,2,\ldots,2)$

TL;DR

This paper develops an analytic method to evaluate specific classes of Euler sums, multiple zeta values, and multiple t-values, and proposes a conjecture for a closed form of a particular multiple t-value.

Contribution

It introduces an analytic approach for evaluating odd Euler sums, multiple zeta values, and multiple t-values, and conjectures a closed form for a specific multiple t-value.

Findings

Analytic evaluation method for odd Euler sums and multiple zeta/t-values.

Conjectured closed form for the multiple t-value t(2,...,2,1).

Enhanced understanding of the structure of these special sums.

Abstract

We construct an analytic approach to evaluate odd Euler sums, multiple zeta value $\zeta(3,2,\ldots,2)$ and multiple $t$-value $t\left(3,2,\ldots,2\right)$. Moreover, we also conjecture a closed expression for multiple $t$-value $t\left(2,\ldots,2,1\right)$.