Discovering New Recurrence Relations for Stirling Numbers: Leveraging a Poisson Expectation Identity for Higher-Order Moments

This paper establishes two novel recurrence relations for Stirling numbers of the second kind—an L recurrence and a vertical recurrence—discovered through a probabilistic analysis of Poisson higher-order origin moments. While the link between these moments and Stirling numbers is known, our derivation via a specific expectation identity provides a clear and efficient pathway to their computation, circumventing the need for infinite series. The primary theoretical contribution is the proof of these previously undocumented combinatorial recurrences, which are of independent mathematical interest. Furthermore, we demonstrate the severe practical inadequacy of high-order sample moments as estimators, highlighting the necessity of our analytical approach to obtaining reliable estimates in applied fields.