{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T15:46:22Z","timestamp":1781711182350,"version":"3.54.5"},"reference-count":21,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,16]],"date-time":"2022-10-16T00:00:00Z","timestamp":1665878400000},"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 consider the degenerate forms of the Catalan\u2013Daehee polynomials and numbers by the Volkenborn integrals and obtain diverse explicit expressions and formulas. Moreover, we show the expressions of the degenerate Catalan\u2013Daehee numbers in terms of \u03bb-Daehee numbers, Stirling numbers of the first kind and Bernoulli polynomials, and we also obtain a relation covering the Bernoulli numbers, the degenerate Catalan\u2013Daehee numbers and Stirling numbers of the second kind. In addition, we prove an implicit summation formula and a symmetric identity, and we derive an explicit expression for the degenerate Catalan\u2013Daehee polynomials including the Stirling numbers of the first kind and Bernoulli polynomials.<\/jats:p>","DOI":"10.3390\/sym14102169","type":"journal-article","created":{"date-parts":[[2022,10,17]],"date-time":"2022-10-17T05:08:02Z","timestamp":1665983282000},"page":"2169","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Note on Degenerate Catalan-Daehee Numbers and Polynomials"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","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":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5567-7444","authenticated-orcid":false,"given":"Maryam Salem","family":"Alatawi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5717-1199","authenticated-orcid":false,"given":"Ugur","family":"Duran","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences of Engineering, Faculty of Engineering and Natural Sciences, \u0130skenderun Technical University, Hatay 31200, T\u00fcrkiye"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,16]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Explicit expressions for Catalan-Daehee numbers","volume":"20","author":"Dolgy","year":"2017","journal-title":"Proc. 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