A generalization of the duality and sum formulas on the multiple zeta values (Q1284197)

The author generalizes a duality theorem of \textit{D. Zagier} [Prog. Math. 120, 497--512 (1994; Zbl 0822.11001)] on multiple zeta values from which several results of \textit{M. Hoffman} [Pac. J. Math. 152, 275--290 (1992; Zbl 0763.11037)] and others are deduced as special cases. Another application is the evaluation of the integral \[ \xi_k(s)= {1\over\Gamma (s)} \int^\infty_0{t^{s-1} \over e^t-1} Li_k(1-e^{-t})\, dt \] for positive integer values of \(s\), where \(Li_k(z)\) denotes the \(k\)th polylogarithm \(\sum^\infty_{m=0} m^{-k}z^m\).