Generalization of Spivey's recurrence relation

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.