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 [Formula: see text]-element set is called the [Formula: see text]th derangement number. Recently, the degenerate derangement numbers and polynomials have been studied as degenerate versions. Let [Formula: see text] 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 [Formula: see text]. In addition, we consider the probabilistic degenerate [Formula: see text]-derangement numbers associated with [Formula: see text] and the probabilistic degenerate derangement polynomials of the second kind associated with [Formula: see text]. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers.