{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T08:07:47Z","timestamp":1778314067742,"version":"3.51.4"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T00:00:00Z","timestamp":1585785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In recent years, many mathematicians studied various degenerate versions of some special polynomials for which quite a few interesting results were discovered. In this paper, we introduce the type 2 degenerate Bernoulli polynomials of the second kind and their higher-order analogues, and study some identities and expressions for these polynomials. Specifically, we obtain a relation between the type 2 degenerate Bernoulli polynomials of the second and the degenerate Bernoulli polynomials of the second, an identity involving higher-order analogues of those polynomials and the degenerate Stirling numbers of the second kind, and an expression of higher-order analogues of those polynomials in terms of the higher-order type 2 degenerate Bernoulli polynomials and the degenerate Stirling numbers of the first kind.<\/jats:p>","DOI":"10.3390\/sym12040510","type":"journal-article","created":{"date-parts":[[2020,4,2]],"date-time":"2020-04-02T13:39:34Z","timestamp":1585834774000},"page":"510","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Some Identities on Type 2 Degenerate Bernoulli Polynomials of the Second Kind"],"prefix":"10.3390","volume":"12","author":[{"given":"Taekyun","family":"Kim","sequence":"first","affiliation":[{"name":"School of Sciences, Xian Technological University, Xi\u2019an 710021, China"},{"name":"Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lee-Chae","family":"Jang","sequence":"additional","affiliation":[{"name":"Graduate School of Education, Konkuk University, Seoul 05029, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9599-7015","authenticated-orcid":false,"given":"Dae San","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sogang University, Seoul 121-742, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Han Young","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kwangwoon University, Seoul 139-701, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,2]]},"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":"28","DOI":"10.1007\/BF01900520","article-title":"A degenerate Staudt-Clausen theorem","volume":"7","author":"Carlitz","year":"1956","journal-title":"Arch. 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RACSAM"},{"key":"ref_21","first-page":"2","article-title":"Degenerate poly-Bernoulli numbers and polynomials","volume":"31","author":"Kim","year":"2020","journal-title":"Informatica"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/510\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:14:33Z","timestamp":1760174073000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/4\/510"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,2]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040510"],"URL":"https:\/\/doi.org\/10.3390\/sym12040510","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,2]]}}}