Generalization of Spivey's recurrence relation

TL;DR

This paper generalizes Spivey's recurrence relation for Bell numbers to probabilistic r-Bell polynomials associated with a random variable Y, extending the combinatorial and probabilistic understanding of these polynomials.

Contribution

It introduces a generalized recurrence relation for probabilistic r-Bell polynomials linked to a random variable Y, broadening the scope of Bell number relations.

Findings

Derived a new recurrence relation for probabilistic r-Bell polynomials

Extended Spivey's relation from Bell numbers to probabilistic polynomials

Established conditions for the existence of the moment generating function of Y

Abstract

In 2008, Spivey found a recurrence relation for the Bell numbers. We consider the probabilistic r-Bell polynomials associated with which are a probabilistic extension of the r-Bell polynomials. Here Y is a random variable whose moment generating function exists in some neighborhood of the origin . The aim of this paper is to generalize the relation for the Bell numbers to that for the probabilistic r-Bell polynomials associated with Y.