A note on derangement polynomials and degenerate derangement polynomials

Abstract

In 1708, Pierre Remonde de Motmort introduced the problem of counting derangement for the first time. A derangement is a permutation that has no fixed points. Recently, many researchers have studied the derangement polynomials and T. Kim introduced the degenerate derangement polynomials and investigated some identities of those polynomials. In, Jang-Kim-Kim-Lee introduced some identities involving derangement polynomials and numbers and moments of gamma random variables. In this paper, we study some identities and properties of the derangement polynomials and degenerate derangement polynomials and investigate the zeros of derangement polynomials. Moreover, we investigate the numerical pattern of the roots of the polynomials Dn,λ(x) varying the degree of polynomials from 1 to 40.