Inverse rising factorial moments Formula: see text are fundamental quantities for positive integer-valued random variables, with broad applications in probability and combinatorics. Despite their importance, there is a lack of explicit, closed-form algebraic identities connecting Formula: see text to established combinatorial quantities and a limited framework for their multivariate extension. Addressing these gaps, this study investigates Formula: see text to derive more computationally efficient alternatives. We establish both a novel finite sum identity and an infinite series expression for Formula: see text. Furthermore, we provide a closed-form expression for the geometric random variable and extend these results to multivariate random variables. Finally, we extend the concept of Formula: see text to positive real-valued continuous random variables, deriving a finite sum identity for the beta random variable.
Li et al. (Tue,) studied this question.