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: $$\sum _{k=0}^n{2}^{n-k}s(n,k)B_k=\sum _{k=0}^{n}b(n,k)\frac{(-1{)}^{k}k!}{k+1}\mathrm{.}$$ 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.