Some identities on derangement and degenerate derangement polynomials

TL;DR

This paper introduces derangement polynomials and degenerate derangement polynomials, exploring their properties, recurrence relations, and identities, thereby extending classical derangement concepts in combinatorics.

Contribution

It presents new polynomial families related to derangements, along with their properties and identities, expanding the combinatorial framework of derangement numbers.

Findings

Derived recurrence relations for derangement polynomials

Established identities connecting these polynomials to special numbers

Extended classical derangement concepts to polynomial forms

Abstract

In combinatorics, a derangement is a permutation that has no fixed points. The number of derangements of an n-element set is called the n-th derangement number. In this paper, as natural companions to derangement numbers and degenerate versions of the companions we introduce derangement polynomials and degenerate derangement polynomials. We give some of their properties, recurrence relations and identities for those polynomials which are related to some special numbers and polynomials.