Probabilistic degenerate derangement polynomials

In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangements of an $n$n-element set is called the $n$nth derangement number. Recently, the degenerate derangement numbers and polynomials have been studied as degenerate versions. Let $Y$Y be a random variable whose moment generating function exists in a neighbourhood of the origin. In this paper, we study probabilistic extension of the degenerate derangement numbers and polynomials, namely the probabilistic degenerate derangement numbers and polynomials associated with $Y$Y. In addition, we consider the probabilistic degenerate $r$r-derangement numbers associated with $Y$Y and the probabilistic degenerate derangement polynomials of the second kind associated with $Y$Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers.