Explicit formulas of sums involving harmonic numbers and Stirling numbers

Abstract

In this paper, we study the explicit formulas of some Euler sums and Stirling sums, which are infinite series on harmonic numbers and Stirling numbers, respectively. As a result, we show that some Stirling sums are expressible in terms of special integrals and alternating multiple zeta values, and all the Euler sums involving negative powers of two are expressible in terms of multiple polylogarithms, hence in terms of unit-exponent alternating multiple zeta values. Some special cases are discussed, and some identities on Euler sums and alternating MZVs are obtained, including a conjectural one due to Borwein et al. Moreover, the Maple program based on the explicit formula is developed, so that the Euler sums involving negative powers of two and of weight can be computed automatically.