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.

中文翻译：

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涉及调和数和斯特林数的和的显式公式

在本文中，我们研究了一些欧拉和和斯特林和的显式公式，它们分别是调和数和斯特林数的无限级数。结果，我们证明了一些斯特林和可以用特殊积分和交替多个 zeta 值表示，并且所有涉及 2 的负幂的欧拉和都可以用多个多对数表示，因此用单位指数交替倍数表示zeta 值。讨论了一些特殊情况，并获得了欧拉和和交替 MZV 的一些恒等式，其中包括 Borwein 等人的推测。此外，开发了基于显式公式的Maple程序，可以自动计算涉及2和权重的负幂的欧拉和。