Generalization of Spivey’s Recurrence Relation

Abstract

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 $$Y$$ .

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