Generalization of Spivey’s Recurrence Relation

In 2008, Spivey found a recurrence relation for the Bell numbers \(\phi_{n}\). We consider the probabilistic \(r\)-Bell polynomials associated with \(Y\), \(\phi_{n,r}^{Y}(x)\), 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 and \(\phi_{n}=\phi_{n,0}^{1}(1)\). The aim of this paper is to generalize the relation for the Bell numbers to that for the probabilistic \(r\)-Bell polynomials associated with \(Y\).