PROBABILISTIC DERANGEMENT NUMBERS AND POLYNOMIALS

Abstract

Let Y be a random variable such that the moment generating function
of Y exists in a neighborhood of the origin. The aim of this paper is to
study probabilistic versions of the derangement polynomials, the derangement polynomials of type 2 and the r-derangement numbers, namely the probabilistic derangement polynomials associated with Y, the probabilistic derangement polynomials of type 2 associated with Y and the probabilistic r-derangement numbers associated Y, respectively. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers. In addition, we consider the special case that Y is the gamma random variable with parameters α,β > 0.