{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:39:58Z","timestamp":1760240398736,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T00:00:00Z","timestamp":1558915200000},"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":["No. 2019R1C1C1003869"],"award-info":[{"award-number":["No. 2019R1C1C1003869"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Here we would like to introduce the extended r-central incomplete and complete Bell polynomials, as multivariate versions of the recently studied extended r-central factorial numbers of the second kind and the extended r-central Bell polynomials, and also as multivariate versions of the r- Stirling numbers of the second kind and the extended r-Bell polynomials. In this paper, we study several properties, some identities and various explicit formulas about these polynomials and their connections as well.<\/jats:p>","DOI":"10.3390\/sym11050724","type":"journal-article","created":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T11:19:27Z","timestamp":1558955967000},"page":"724","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On r-Central Incomplete and Complete Bell 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 04107, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Han Young","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kwangwoon University, Seoul 01897, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dojin","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Pusan National University, Busan 46241, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taekyun","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kwangwoon University, Seoul 01897, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,27]]},"reference":[{"key":"ref_1","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. arXiv.","DOI":"10.1186\/s13662-019-2195-0"},{"key":"ref_2","unstructured":"Riordan, J. (1968). Combinatorial Identities, John Wiley & Sons, Inc."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kim, T., Kim, D.S., Jang, G.-W., and Kwon, J. (2019). Extended central factorial polynomials of the second kind. Adv. Differ. Equ., 24.","DOI":"10.1186\/s13662-019-1963-1"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kim, T., and Kim, D.S. (2019). Degenerate central Bell numbers and polynomials. Rev. Real Acad. Clenc. Exactas Fis. Nat. Ser. A Mat., 1\u20137.","DOI":"10.1007\/s13398-019-00637-0"},{"key":"ref_5","first-page":"154","article-title":"Some identities involving the Euler and the central factorinal numbers","volume":"36","author":"Zhang","year":"1998","journal-title":"Fibonacci Quart."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1080\/01630568908816313","article-title":"Central factorial numbers; their main properties and some applications","volume":"10","author":"Butzer","year":"1989","journal-title":"Numer. Funct. Anal. Optim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"94","DOI":"10.4153\/CJM-1963-010-8","article-title":"The divided central differences of zero","volume":"15","author":"Carlitz","year":"1963","journal-title":"Canad. J. Math."},{"key":"ref_8","first-page":"66","article-title":"Some remarks on the Bell numbers","volume":"18","author":"Carlitz","year":"1980","journal-title":"Fibonacci Quart."},{"key":"ref_9","first-page":"451","article-title":"Central factorial numbers and related expansions","volume":"19","author":"Charalambides","year":"1981","journal-title":"Fibonacci Quart."},{"key":"ref_10","first-page":"575","article-title":"A note on central factorial numbers","volume":"21","author":"Kim","year":"2018","journal-title":"Proc. Jangjeon Math. Soc."},{"key":"ref_11","first-page":"289","article-title":"A note on central Bell numbers and polynomials","volume":"27","author":"Kim","year":"2019","journal-title":"Adv. Stud. Contemp. Math. (Kyungshang)"},{"key":"ref_12","unstructured":"Belbachir, H., and Djemmada, Y. (2018). On central Fubini-like numbers and polynomials. arXiv."},{"key":"ref_13","unstructured":"Krzywonos, N., and Alayont, F. (2009). Rook polynomials in higher dimensions. Stud. Summer Sch., 29, Available online: https:\/\/scholarworks.gvsu.edu\/sss\/29\/."},{"key":"ref_14","first-page":"9","article-title":"On (q,r,w)-Stirling numbers of the second kind","volume":"9","author":"Duran","year":"2018","journal-title":"J. Inequal. Spec. Funct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1134\/S1061920818010041","article-title":"Degenerate r-Stirling numbers and r-Bell polynomials","volume":"25","author":"Kim","year":"2018","journal-title":"Russ. J. Math. Phys."},{"key":"ref_16","first-page":"127","article-title":"Degenerate Cauchy numbers and polynomials of the fourth kind","volume":"28","author":"Pyo","year":"2018","journal-title":"Adv. Stud. Contemp. Math. (Kyungshang)"},{"key":"ref_17","unstructured":"Roman, S. (1984). The umbral calculus. Pure and Applied Mathematics, Harcourt Brace Jovanovich."},{"key":"ref_18","first-page":"127","article-title":"Identities and relations related to combinatorial numbers and polynomials","volume":"20","author":"Simsek","year":"2017","journal-title":"Proc. Jangjeon Math. Soc."},{"key":"ref_19","first-page":"199","article-title":"Identities on the Changhee numbers and Apostol-type Daehee polynomials","volume":"27","author":"Simsek","year":"2017","journal-title":"Adv. Stud. Contemp. Math. (Kyungshang)"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1167","DOI":"10.1007\/s13370-017-0510-z","article-title":"The partial r-Bell polynomials","volume":"28","author":"Mihoubi","year":"2017","journal-title":"Afr. Mat."},{"key":"ref_21","unstructured":"Kim, T., Kim, D.S., and Jang, G.-W. (2019). On central complete and incomplete Bell polynomials I. Symmetry, 11."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/724\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:55:30Z","timestamp":1760187330000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/724"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,27]]},"references-count":21,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050724"],"URL":"https:\/\/doi.org\/10.3390\/sym11050724","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,5,27]]}}}