{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T19:13:20Z","timestamp":1780514000142,"version":"3.54.1"},"reference-count":31,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["NRF-2018R1D1A1B07049584"],"award-info":[{"award-number":["NRF-2018R1D1A1B07049584"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, Kim-Kim (J. Math. Anal. Appl. (2021), Vol. 493(1), 124521) introduced the \u03bb-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\u2208N) 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.<\/jats:p>","DOI":"10.3390\/sym13020176","type":"journal-article","created":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T12:28:31Z","timestamp":1611577711000},"page":"176","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Some Identities of the Degenerate Higher Order Derangement Polynomials and Numbers"],"prefix":"10.3390","volume":"13","author":[{"given":"Hye Kyung","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics Education, Daegu Catholic University, Gyeongsan 38430, Korea"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,22]]},"reference":[{"key":"ref_1","first-page":"51","article-title":"Degenerate Stirling, Bernoulli and Eulerian numbers","volume":"15","author":"Carlitz","year":"1979","journal-title":"Utilitas Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1548","DOI":"10.1137\/070703697","article-title":"The two-coloring number and degenerate coloring of planer graphs","volume":"23","author":"Kierstead","year":"2009","journal-title":"Siam J. 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