An Operator Theoretic Derivation of a Bernoulli Stirling Identity: A Novel Use of the Euler Operator and Integration

Abstract: We present a novel proof of the classical identity relating Bernoulli numbers $B_n$ and Stirling numbers of the second kind $S(n,k)$, given by [ B_n = \sum_{k=1}^n S(n,k)\frac{(-1)^k k!}{k+1}. ] Unlike traditional derivations based on generating functions or purely combinatorial arguments, our method leverages the Euler operator $\vartheta = t \frac{d}{dt}$ and its algebraic action on analytic functions. By applying this operator repeatedly to a suitably chosen function and integrating both sides, we derive the identity in a way that reveals a deeper operator-theoretic structure underlying the formula. This approach bridges discrete combinatorics and differential operators, offering a fresh perspective on a classical result.