{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:41:44Z","timestamp":1760240504097,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,13]],"date-time":"2019-07-13T00:00:00Z","timestamp":1562976000000},"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":["2017R1E1A1A03070882"],"award-info":[{"award-number":["2017R1E1A1A03070882"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The purpose of this paper is to introduce and study type 2 degenerate q-Bernoulli polynomials and numbers by virtue of the bosonic p-adic q-integrals. The obtained results are, among other things, several expressions for those polynomials, identities involving those numbers, identities regarding Carlitz\u2019s q-Bernoulli numbers, identities concerning degenerate q-Bernoulli numbers, and the representations of the fully degenerate type 2 Bernoulli numbers in terms of moments of certain random variables, created from random variables with Laplace distributions. It is expected that, as was done in the case of type 2 degenerate Bernoulli polynomials and numbers, we will be able to find some identities of symmetry for those polynomials and numbers.<\/jats:p>","DOI":"10.3390\/sym11070914","type":"journal-article","created":{"date-parts":[[2019,7,15]],"date-time":"2019-07-15T04:55:27Z","timestamp":1563166527000},"page":"914","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Note on Type 2 Degenerate q-Bernoulli Polynomials"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9599-7015","authenticated-orcid":false,"given":"Dae San","family":"Kim","sequence":"first","affiliation":[{"name":"Department of Mathematics, Sogang University, Seoul 121-742, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dmitry V.","family":"Dolgy","sequence":"additional","affiliation":[{"name":"Kwangwoon Institute for Advanced Studies, Kwangwoon University, Seoul 139-701, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jongkyum","family":"Kwon","sequence":"additional","affiliation":[{"name":"Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Gyeongsangnamdo 52828, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taekyun","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":[[2019,7,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kim, D.S., Kim, H.Y., Kim, D., and Kim, T. (2019). Identities of symmetry for type 2 Bernoulli and Euler polynomials. Symmetry, 11.","DOI":"10.3390\/sym11050613"},{"key":"ref_2","first-page":"51","article-title":"Degenerate Stirling, Bernoulli and Eulerian numbers","volume":"15","author":"Carlitz","year":"1979","journal-title":"Utilitas Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1186\/s13662-019-2129-x","article-title":"Some identities of special numbers and polynomials arising from p-adic integrals on Zp","volume":"2019","author":"Kim","year":"2019","journal-title":"Adv. Differ. Equ."},{"key":"ref_4","first-page":"288","article-title":"q-Volkenborn integration","volume":"9","author":"Kim","year":"2002","journal-title":"Russ. J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kim, D.S., Kim, T., Kim, H.Y., and Kwon, J. (2019). A note on type 2 q-Bernoulli and type 2 q-Euler polynomials. J. Inequal. Appl., accepted.","DOI":"10.1186\/s13660-019-2131-6"},{"key":"ref_6","first-page":"399","article-title":"A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials","volume":"22","author":"Araci","year":"2012","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1215\/S0012-7094-58-02532-8","article-title":"Expansions of q-Bernoulli numbers","volume":"25","author":"Carlitz","year":"1958","journal-title":"Duke Math. J."},{"key":"ref_8","first-page":"147","article-title":"A note on type 2 degenerate Euler and Bernoulli polynomials","volume":"29","author":"Jang","year":"2019","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Kim, D.S., Dolgy, D.V., Kim, D., and Kim, T. (2019). Some identities on r-central factorial numbers and r-central Bell polynomials. Adv. Differ. Equ., accepted.","DOI":"10.1186\/s13662-019-2195-0"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1090\/conm\/596\/11902","article-title":"The p-adic q-distributions, Advances in ultrametric analysis","volume":"596","author":"Diarra","year":"2013","journal-title":"Contemp. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1134\/S1061920817020091","article-title":"Degenerate Laplace transform and degenerate gamma function","volume":"24","author":"Kim","year":"2017","journal-title":"Russ. J. Math. Phys."},{"key":"ref_12","first-page":"393","article-title":"A note on degenerate Stirling numbers of the first kind","volume":"21","author":"Kim","year":"2018","journal-title":"Proc. Jangjeon Math. Soc."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.4134\/BKMS.b150583","article-title":"On degenerate q-Bernoulli polynomials","volume":"53","author":"Kim","year":"2016","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Schikhof, W.H. (1984). Ultrametric Calculus. An Introduction to p-Adic Analysis, Cambridge University Press. Cambridge Studies in Advanced Mathematics, 4.","DOI":"10.1017\/CBO9780511623844"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/914\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:05:27Z","timestamp":1760187927000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/914"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,13]]},"references-count":14,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7]]}},"alternative-id":["sym11070914"],"URL":"https:\/\/doi.org\/10.3390\/sym11070914","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,7,13]]}}}