Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers

Abstract

Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. In addition, Kim et al. (arXiv:2011.08535v1 17 November 2020) studied the degenerate derangement polynomials and numbers, and investigated some properties of those polynomials without using degenerate umbral calculus. In this paper, the y the degenerate derangement polynomials of order s (s∈N) and give a combinatorial meaning about higher order derangement numbers. In addition, the author gives some interesting identities related to the degenerate derangement polynomials of order s and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derive the inversion formulas of these identities.

Keywords

• derangement numbers and polynomials
• degenerate derangement numbers and polynomials
• Lah–Bell numbers and polynomials
• the degenerate Sheffer sequence
• the degenerate Bernoulli (Euler) polynomials
• the degenerate Frobenius–Euler polynomials