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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.15380 |
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| _version_ | 1866910054776569856 |
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| author | Kikuchi, Tomoko Nakasuji, Maki |
| author_facet | Kikuchi, Tomoko Nakasuji, Maki |
| contents | The classical Bernoulli numbers $B_m$ can be expressed using Stirling numbers of the second kind, and M. Kaneko extended this framework by defining poly-Bernoulli numbers ${\mathbb B}_m^{(k)}$, for which explicit formulas using the Stirling numbers of the second kind and duality relations were obtained. Later, Kaneko and H. Tsumura introduced multi-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, \ldots, m_r}^{(k_1, \ldots, k_r)}$ using the multiple polylogarithm and reached their duality properties via an associated $η$-function. Explicit formulas for double-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, m_2}^{(k_1, k_2)}$ were obtained by Y. Baba, M. Nakasuji, and M. Sakata. In this article, we extend these results to general multi-indexed poly-Bernoulli numbers and use it to give an alternative proof of the duality of multi-indexed poly-Bernoulli numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15380 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit formula for multi-indexed poly-Bernoulli numbers Kikuchi, Tomoko Nakasuji, Maki Number Theory 11M68 The classical Bernoulli numbers $B_m$ can be expressed using Stirling numbers of the second kind, and M. Kaneko extended this framework by defining poly-Bernoulli numbers ${\mathbb B}_m^{(k)}$, for which explicit formulas using the Stirling numbers of the second kind and duality relations were obtained. Later, Kaneko and H. Tsumura introduced multi-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, \ldots, m_r}^{(k_1, \ldots, k_r)}$ using the multiple polylogarithm and reached their duality properties via an associated $η$-function. Explicit formulas for double-indexed poly-Bernoulli numbers ${\mathbb B}_{m_1, m_2}^{(k_1, k_2)}$ were obtained by Y. Baba, M. Nakasuji, and M. Sakata. In this article, we extend these results to general multi-indexed poly-Bernoulli numbers and use it to give an alternative proof of the duality of multi-indexed poly-Bernoulli numbers. |
| title | Explicit formula for multi-indexed poly-Bernoulli numbers |
| topic | Number Theory 11M68 |
| url | https://arxiv.org/abs/2603.15380 |