Work

Some representations for inverse rising factorial moments

Year: 2026

Type: article

Abstract: 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.

Source: Mathematical and Computer Modelling of Dynamical Systems

DOI: 10.1080/13873954.2026.2667028

Authors: Xiaoxue Li, Yuankui Ma, Taekyun Kim, Dae San Kim

Institutions: Xi'an Technological University, Kwangwoon University, Sogang University

Language: English

Topic: Probability and Statistical Research

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $1,533

Funder: Natural Science Basic Research Program of Shaanxi Province