Moment Generating Stirling Numbers and Applications

TL;DR

This paper introduces moment generating Stirling numbers, explores their properties, and applies them to compute moments of various distributions and processes in closed form, enhancing analytical tools in combinatorics and probability.

Contribution

It defines and studies properties of moment generating Stirling numbers, a special case of generalized Stirling numbers, and applies them to compute moments of multiple distributions without recursion.

Findings

Closed-form formulas for moments and central moments of several distributions.

New combinatorial identities involving moment generating Stirling numbers.

Relationship established between these numbers and Markov renewal processes.

Abstract

In this paper, we investigate certain combinatorial numbers, the \textit{moment generating Stirling numbers}. They are a special case of Hsu's generalized Stirling numbers and satisfy many more properties and combinatorial identities than are known in the general case. As application, we provide the computation of the moments and central moments of the phase type distribution, the recurrence time in Markov chains, the geometric distribution, the negative binomial distribution and of a class of distributions generalizing the negative binomial distribution. All computations can be performed in closed form without recursion. We also present the relationship to the Markov renewal process.