A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916948539867136 |
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| author | Benmoussa, Abdelhay |
| author_facet | Benmoussa, Abdelhay |
| contents | We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: \[ \sum_{k=0}^{n} 2^{\,n-k}\,s(n,k)\,B_k \;=\; \sum_{k=0}^{n} b(n,k)\,\frac{(-1)^k\,k!}{k+1}. \] This parallels the classical Stirling--Bernoulli relation \[ B_n = \sum_{k=0}^{n} S(n,k)\,\frac{(-1)^k\,k!}{k+1}, \] replacing $S(n,k)$ with $s(n,k)$ and $b(n,k)$, and thus revealing a new structural connection among these families of numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_22819 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind Benmoussa, Abdelhay General Mathematics 11B68, 05A18, 11B73, 33C10 We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: \[ \sum_{k=0}^{n} 2^{\,n-k}\,s(n,k)\,B_k \;=\; \sum_{k=0}^{n} b(n,k)\,\frac{(-1)^k\,k!}{k+1}. \] This parallels the classical Stirling--Bernoulli relation \[ B_n = \sum_{k=0}^{n} S(n,k)\,\frac{(-1)^k\,k!}{k+1}, \] replacing $S(n,k)$ with $s(n,k)$ and $b(n,k)$, and thus revealing a new structural connection among these families of numbers. |
| title | A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind |
| topic | General Mathematics 11B68, 05A18, 11B73, 33C10 |
| url | https://arxiv.org/abs/2505.22819 |