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arXiv · 2505.22819

A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind

Abdelhay Benmoussa
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Abstract

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.

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Abdelhay Benmoussa. 2025-05-28. A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind. https://arxiv.org/abs/2505.22819

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