{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T17:33:00Z","timestamp":1768584780048,"version":"3.49.0"},"reference-count":27,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,25]],"date-time":"2022-12-25T00:00:00Z","timestamp":1671926400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating function methods and we illustrate our results with some examples.<\/jats:p>","DOI":"10.3390\/axioms12010021","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T03:03:31Z","timestamp":1672110211000},"page":"21","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["New Classes of Degenerate Unified Polynomials"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5614-4279","authenticated-orcid":false,"given":"Daniel","family":"Bedoya","sequence":"first","affiliation":[{"name":"Departamento de Ciencias B\u00e1sicas, Universidad Metropolitana, Barranquilla 00928-1345, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Clemente","family":"Cesarano","sequence":"additional","affiliation":[{"name":"Section of Mathematics, Universit\u00e1 Telematica Internazionale Uninettuno, 00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9613-9469","authenticated-orcid":false,"given":"Stiven","family":"D\u00edaz","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Barranquilla 00928-1345, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4675-0221","authenticated-orcid":false,"given":"William","family":"Ram\u00edrez","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias Naturales y Exactas, Universidad de la Costa, Barranquilla 00928-1345, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,25]]},"reference":[{"key":"ref_1","unstructured":"Abramowitz, M., and Stegun, I. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."},{"key":"ref_2","unstructured":"Apostol, T.M. (1998). Introduction to Analytic Number Theory, Springer Science & Business Media."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1063\/1.4822863","article-title":"Concrete mathematics: A foundation for computer science","volume":"3","author":"Graham","year":"1989","journal-title":"Comput. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s00025-014-0430-2","article-title":"About Extensions Of Generalized Apostol-type polynomials","volume":"68","author":"Quintana","year":"2015","journal-title":"Results Math."},{"key":"ref_5","first-page":"2315","article-title":"A further generalization of the Bernoulli polynomials and on the 2D\u2013Bernoulli polynomials Bn2(x,y)","volume":"47","author":"Kurt","year":"2010","journal-title":"Appl. Math. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"604","DOI":"10.37394\/23206.2022.21.69","article-title":"New results for degenerated generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials","volume":"21","author":"Cesarano","year":"2022","journal-title":"WSEAS Trans. Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ram\u00edrez, W., Castilla, L., and Urieles, A. (2018). An extended generalized-extensions for the Apostol Type polynomial. Abstr. Appl. Anal., 2018.","DOI":"10.1155\/2018\/2937950"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1377","DOI":"10.1016\/j.camwa.2006.02.003","article-title":"Explicit formulas for the Norlund polynomials Bn(x) and bn(x)","volume":"51","author":"Srivastava","year":"2006","journal-title":"Comput. Math. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1080\/00150517.1992.12429381","article-title":"Negative order Genocchi polynomials","volume":"30","author":"Horadam","year":"1992","journal-title":"Fibonacci Quart."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"161","DOI":"10.2140\/pjm.1951.1.161","article-title":"On the Lerch zeta function","volume":"1","author":"Apostol","year":"1951","journal-title":"Pac. J. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1016\/j.jmaa.2005.01.020","article-title":"Some generalizations of the Apostol\u2013Bernoulli and Apostol\u2013Euler polynomials","volume":"308","author":"Luo","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"917","DOI":"10.11650\/twjm\/1500403883","article-title":"Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions","volume":"10","author":"Luo","year":"2006","journal-title":"Taiwan. J. Math."},{"key":"ref_13","first-page":"390","article-title":"Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials","volume":"5","author":"Srivastava","year":"2011","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Belbachir, H., Djemmada, Y., and Hadj-Brahim, S. (2022). Unified Bernoulli-Euler polynomials of Apostol type. Indian J. Pure Appl. Math., 1\u20138.","DOI":"10.1007\/s13226-022-00232-x"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"411","DOI":"10.46793\/KgJMat2503.411R","article-title":"New results parametric Apostol-type Frobenius-Euler polynomials and their matrix approach","volume":"49","author":"Bedoya","year":"2025","journal-title":"Kragujev. J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-018-1526-x","article-title":"Construction of fourier expansion of Apostol Frobenius-Euler polynomials and its applications","volume":"2018","author":"Araci","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Andrews, L.C. (1998). Special Functions of Mathematics for Engineers, SPIE Press.","DOI":"10.1093\/oso\/9780198565581.001.0001"},{"key":"ref_18","first-page":"1","article-title":"A new class of degenerate Apostol\u2013type Hermite polynomials and applications","volume":"15","author":"Cesarano","year":"2022","journal-title":"Dolomites Res. Notes Approx."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Hwang, K.W., and Ryoo, C.S. (2020). Some identities involving two-variable partially degenerate Hermite polynomials induced from differential equations and structure of their roots. Mathematics, 8.","DOI":"10.3390\/math8040632"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1007\/s40590-018-0220-z","article-title":"On degenerate Apostol-type polynomials and applications","volume":"25","author":"Khan","year":"2019","journal-title":"Bol. Soc. Matem\u00e1tica Mex."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1007\/BF01900520","article-title":"A degenerate Staudt\u2013Clausen theorem","volume":"7","author":"Carlitz","year":"1956","journal-title":"Arch. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2086","DOI":"10.22436\/jnsa.009.05.14","article-title":"Identities involving degenerate Euler numbers and polynomials arising from nonlinear differential equations","volume":"9","author":"Kim","year":"2016","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"569","DOI":"10.4134\/BKMS.2016.53.2.569","article-title":"Some identities of degenerate Genocchi polynomials","volume":"53","author":"Lim","year":"2016","journal-title":"Bull. Korean Math. Soc."},{"key":"ref_24","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, Academic Press, Inc.. [7th ed.]."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Bloch, E.D. (2011). The Real Numbers and Real Analysis, Springer.","DOI":"10.1007\/978-0-387-72177-4"},{"key":"ref_26","unstructured":"Aigner, M. (2006). Diskrete Mathematik, Friedr. Vieweg & Sohn. [6th ed.]."},{"key":"ref_27","first-page":"319","article-title":"A note on degenerate Stirling polynomials of the second kind","volume":"20","author":"Kim","year":"2017","journal-title":"Proc. Jangjeon Math. Soc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:50:48Z","timestamp":1760147448000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/1\/21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,25]]},"references-count":27,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["axioms12010021"],"URL":"https:\/\/doi.org\/10.3390\/axioms12010021","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,25]]}}}