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

We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: \[ ∑ₖ₌₀ⁿ 2\,n-k\,s(n,k)\,B_k \;=\; ∑ₖ₌₀ⁿ b(n,k)\,(-1)^k\,k!/k+1. \] This parallels the classical Stirling--Bernoulli relation \[ B_n = ∑ₖ₌₀ⁿ S(n,k)\,(-1)^k\,k!/k+1, \] replacing $S(n,k)$ with $s(n,k)$ and $b(n,k)$, and thus revealing a new structural connection among these families of numbers.