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<p>In this paper, we establish a general transformation between <i>r</i>-Stirling series and binomial coefficient series. Combining this with a summation lemma on alternating multiple series, we show that in some cases, the two series in this transformation are expressible in terms of unit-exponent alternating multiple zeta values (AMZVs). Moreover, by specifying the parameters, the AMZV expressions for some parametric Apéry-type series are obtained. As applications, the evaluations of some special series involving central binomial coefficients, Stirling numbers, <i>r</i>-Stirling numbers, and hyperharmonic numbers, as well as some binomial coefficient identities are presented, including some known ones in the literature and some new ones.</p>

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A Transformation Between r-Stirling and Binomial Coefficient Series Via Alternating Multiple Zeta Values

  • 期刊论文

  • 发表日期：

    2026年5月21日

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  • Xin Chen,
  • Weiping Wang

摘要

In this paper, we establish a general transformation between r-Stirling series and binomial coefficient series. Combining this with a summation lemma on alternating multiple series, we show that in some cases, the two series in this transformation are expressible in terms of unit-exponent alternating multiple zeta values (AMZVs). Moreover, by specifying the parameters, the AMZV expressions for some parametric Apéry-type series are obtained. As applications, the evaluations of some special series involving central binomial coefficients, Stirling numbers, r-Stirling numbers, and hyperharmonic numbers, as well as some binomial coefficient identities are presented, including some known ones in the literature and some new ones.