A probabilistic generalization of the Stirling numbers of the second kind

TL;DR

This paper introduces a new probabilistic generalization of Stirling numbers of the second kind linked to random variables with finite moment generating functions, extending classical sum formulas and exploring various mathematical applications.

Contribution

It presents a novel probabilistic framework for Stirling numbers, providing characterizations, examples, and applications to sums of powers and special polynomials.

Findings

New probabilistic generalization of Stirling numbers

Extensions of classical sum formulas for powers

Connections to Bell polynomials, polylogarithms, and Appell polynomials

Abstract

Associated to each random variable $Y$ having a finite moment generating function, we introduce a different generalization of the Stirling numbers of the second kind. Some characterizations and specific examples of such generalized numbers are provided. As far as their applications are concerned, attention is focused in extending in various ways the classical formula for sums of powers on arithmetic progressions. Illustrations involving rising factorials, Bell polynomials, polylogarithms, and a certain class of Appell polynomials, in connection with appropriate random variables $Y$ in each case, are discussed in detail.