New Proofs of the Explicit Formulas of Arakawa--Kaneko Zeta Values and Kaneko--Tsumura $\eta$- and $\psi$- Values

In this paper, we establish some new identities of integrals involving multiple polylogarithm functions and their level two analogues in terms of Hurwitz-type multiple zeta (star) values. Using these identities, we provide new proofs of the explicit formulas of Arakawa–Kaneko zeta values, Kaneko–Tsumura $\eta$- and $\psi$-values, and also give a formula for double $T$-values.