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

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 $t\left(2,\ldots,2,1\right)$.