Some representations for inverse rising factorial moments

Inverse rising factorial moments M−n 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 M−n to established combinatorial quantities and a limited framework for their multivariate extension. Addressing these gaps, this study investigates M−n to derive more computationally efficient alternatives. We establish both a novel finite sum identity and an infinite series expression for M−n. 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 M−n to positive real-valued continuous random variables, deriving a finite sum identity for the beta random variable.