Generalization of Spivey’s Recurrence Relation

In 2008, Spivey found a recurrence relation for the Bell numbers ₍. We consider the probabilistic r -Bell polynomials associated with Y, ₍, ₑ^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 ₍=₍, 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.