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 \(Y\) .