Probabilistic degenerate derangement polynomials

TL;DR

This paper introduces a probabilistic extension of degenerate derangement numbers and polynomials, exploring their properties, identities, and recurrence relations within a probabilistic framework.

Contribution

It presents the first probabilistic generalization of degenerate derangement numbers and polynomials, including associated identities and recurrence relations.

Findings

Derived explicit expressions for probabilistic degenerate derangement numbers.

Established recurrence relations and identities for these probabilistic polynomials.

Extended the concept to probabilistic degenerate r-derangement numbers and second kind polynomials.

Abstract

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 derangement of an n-element set is called the nth derangement number. Recently, the degenerate derangement numbers and polynomials have been studied as degenerate versions. Let Y be a random variable whose moment generating function exists in a neighborhood 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. In addition, we consider the probabilistic degenerate r-derangement numbers associated with Y and the probabilistic degenerate derangement polynomila of the second kind associated with Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers.