In this paper, we study some Euler–Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to $ln\left(2\right)$, zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma function, polygamma function and trigonometric functions. The evaluations of a large number of special Euler–Apéry-type series are presented as examples.

中文翻译：

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交替使用多个zeta值和某些Euler-Apéry类型系列的显式

在本文中，我们研究了一些涉及中心二项式系数和（广义）调和数的Euler-Apéry型序列。特别是，我们通过迭代积分和交替多个zeta值来建立某些系列的优雅显式公式。基于这些公式，我们进一步证明了其他一些级数可以还原为$ln（2）$，zeta值，以及通过考虑与伽马函数，多伽马函数和三角函数有关的轮廓积分来交替使用多个zeta值。举例说明了对大量特殊的Euler-Apéry型系列的评估。