Degenerate Derangement Polynomials and Numbers

Minyoung Ma,Dongkyu Lim

doi:10.3390/fractalfract5030059

Abstract

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.

Highlights

Academic Editors: Praveen Agarwal, Hari Mohan Srivastava, Taekyun Kim and Shaher MomaniReceived: 10 May 2021Accepted: 18 June 2021Published: 22 June 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland.This article is an open access articleA derangement is a permutation with no fixed points
We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ ∈ (−1, 0)
Motivated from the work of Kim et al in [15], in this paper, we have dealt with a new type of degenerate derangement polynomials, which are called the degenerate derangement polynomials of the second kind d∗n,λ ( x )

Summary

Introduction

Academic Editors: Praveen Agarwal, Hari Mohan Srivastava, Taekyun Kim and Shaher Momani. We compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.. By (15) and (16), we obtain the following theorem, which give the expression of the linear combination of the degenerate derangement polynomials of the second kind and the degenerate Stirling numbers of the second kind. This can be compared to those in ([15], Theorem 4). By (28) and (31), we obtain an expression of the linear combination of the degenerate derangement polynomials of the second kind which the coefficients of the degenerate Stirling numbers with −λ. Comparing the coefficients on both sides of (38), we obtain the following theorem

Further Remarks

Conclusions