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

TL;DR

This paper introduces a novel mathematical identity linking Bernoulli, Stirling, and Bessel numbers, revealing a new structural relationship among these classical number families.

Contribution

It establishes a new identity that connects Bernoulli, Stirling (first kind), and Bessel (first kind) numbers, expanding the understanding of their interrelations.

Findings

Derived a new identity linking the three number families.

Revealed structural parallels to classical Stirling--Bernoulli relations.

Enhanced the theoretical framework of special number relationships.

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.