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<p>As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, an explicit formula using the Stirling numbers of the second kind, and a periodicity for negative upper-index have been established. For the multi-indexed poly-Bernoulli numbers generalized by M.&#xa0;Kaneko and H.&#xa0;Tsumura, among such properties only the duality formula has been obtained. In this article, we focus on the double-indexed poly-Bernoulli numbers and show an explicit formula using the Stirling numbers of the second kind, along with a periodicity for the negative upper-index in this specific case. Furthermore, we define a variant of multi-indexed poly-Bernoulli numbers using the star-version of multiple polylogarithms and express this kind of double-indexed poly-Bernoulli numbers in terms of the original multi-indexed poly-Bernoulli numbers. In addition, we discuss the periodicity of it.</p>

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On a Certain Periodicity of Double-Indexed Poly-Bernoulli Numbers

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Yuna Baba,
  • Maki Nakasuji,
  • Mika Sakata

摘要

As properties of poly-Bernoulli numbers, a number of formulas such as the duality formula, an explicit formula using the Stirling numbers of the second kind, and a periodicity for negative upper-index have been established. For the multi-indexed poly-Bernoulli numbers generalized by M. Kaneko and H. Tsumura, among such properties only the duality formula has been obtained. In this article, we focus on the double-indexed poly-Bernoulli numbers and show an explicit formula using the Stirling numbers of the second kind, along with a periodicity for the negative upper-index in this specific case. Furthermore, we define a variant of multi-indexed poly-Bernoulli numbers using the star-version of multiple polylogarithms and express this kind of double-indexed poly-Bernoulli numbers in terms of the original multi-indexed poly-Bernoulli numbers. In addition, we discuss the periodicity of it.