{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T02:23:39Z","timestamp":1775874219263,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,29]],"date-time":"2022-05-29T00:00:00Z","timestamp":1653782400000},"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 this paper, we introduce q-analogues of degenerate Catalan numbers and polynomials with the help of a fermionic p-adic q-integrals on Zp and establish some new connections with the degenerate Stirling numbers of the first and second kinds. Furthermore, we also find a few new identities and results of this type of polynomials and numbers.<\/jats:p>","DOI":"10.3390\/sym14061119","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"1119","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A Note on q-analogue of Degenerate Catalan Numbers Associated with p-adic Integral on Zp"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem A.","family":"Khan","sequence":"first","affiliation":[{"name":"Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box 1664, Al Khobar 31952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,29]]},"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. Math."},{"key":"ref_3","first-page":"651","article-title":"Degenerate Bernoulli numbers and polynomials associated with Hermite polynomials","volume":"33","author":"Haroon","year":"2018","journal-title":"Commun. Korean Math. Soc."},{"key":"ref_4","first-page":"1243","article-title":"Triple symmetric identities for w-Catalan polynomials","volume":"54","author":"Kim","year":"2017","journal-title":"J. Korean Math. Soc."},{"key":"ref_5","first-page":"493","article-title":"A note on Catalan numbers associated with p-adic integral on Zp","volume":"19","author":"Kim","year":"2016","journal-title":"Proc. Jangjeon Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"465","DOI":"10.1134\/S1061920817040057","article-title":"A new approach to Catalan numbers using differential equations","volume":"24","author":"Kim","year":"2017","journal-title":"Russ. J. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.3906\/mat-2008-24","article-title":"New classes of Catalan-type numbers and polynomials with their applications related to p-adic integrals and computational algorithms","volume":"44","author":"Kucukoglu","year":"2020","journal-title":"Turk. J. Math."},{"key":"ref_8","first-page":"279","article-title":"A note on q-analogues of Catalan numbers arising from fermionic p-adic q-integral on Zp","volume":"31","author":"Kim","year":"2021","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Araci, S., Acikgoz, M., and Kilicman, A. (2013). Extended p-adic q-invariant integrals on Zp associated with applications of umbral calculus. Adv. Differ. Equ., 2013.","DOI":"10.1186\/1687-1847-2013-96"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1142\/S1793042118501737","article-title":"On applications for Mahler expansion associated with p-adic q-integrals","volume":"15","author":"Duran","year":"2019","journal-title":"Int. J. Number Theory"},{"key":"ref_11","first-page":"288","article-title":"q-Volkenborn integration","volume":"9","author":"Kim","year":"2002","journal-title":"Russ. J. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"8","DOI":"10.2991\/jnmp.2007.14.1.2","article-title":"An invariant p-adic q-integral associated with q-Euler numbers and polynomials","volume":"14","author":"Cangul","year":"2007","journal-title":"J. Nonlinear Math. Phys."},{"key":"ref_13","first-page":"177","article-title":"Some identities of Catalan-Daehee polynomials arising from umbral calculus","volume":"16","author":"Kim","year":"2017","journal-title":"Appl. Comput. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Khan, W.A. (2022). A note on q-analogues of degenerate Catalan-Daehee numbers and polynomials. J. Math., 2022.","DOI":"10.1155\/2022\/9486880"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2157","DOI":"10.12988\/ijma.2015.57184","article-title":"On degenerate q-Daehee numbers and polynomials","volume":"9","author":"Jeong","year":"2015","journal-title":"Int. J. Math. Anal."},{"key":"ref_16","first-page":"307","article-title":"On the degenerate Cauchy numbers and polynomials","volume":"18","author":"Kim","year":"2015","journal-title":"Proc. Jangjeon Math. Soc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1119\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:20:52Z","timestamp":1760138452000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1119"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,29]]},"references-count":16,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061119"],"URL":"https:\/\/doi.org\/10.3390\/sym14061119","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,29]]}}}