{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T03:14:27Z","timestamp":1769915667565,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,4]],"date-time":"2023-07-04T00:00:00Z","timestamp":1688428800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Research Deanship at the University of Ha\u2019il, Saudi Arabia","award":["RG-21 144"],"award-info":[{"award-number":["RG-21 144"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many properties of special polynomials, such as recurrence relations, sum formulas, and symmetric properties, have been studied in the literature with the help of generating functions and their functional equations. In this study, we define Frobenius\u2013Euler\u2013Genocchi polynomials and investigate some properties by giving many relations and implementations. We first obtain different relations and formulas covering addition formulas, recurrence rules, implicit summation formulas, and relations with the earlier polynomials in the literature. With the help of their generating function, we obtain some new relations, including the Stirling numbers of the first and second kinds. We also obtain some new identities and properties of this type of polynomial. Moreover, using the Fa\u00e0 di Bruno formula and some properties of the Bell polynomials of the second kind, we obtain an explicit formula for the Frobenius\u2013Euler polynomials of order \u03b1. We provide determinantal representations for the ratio of two differentiable functions. We find a recursive relation for the Frobenius\u2013Euler polynomials of order \u03b1. Using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained.<\/jats:p>","DOI":"10.3390\/sym15071358","type":"journal-article","created":{"date-parts":[[2023,7,4]],"date-time":"2023-07-04T01:32:18Z","timestamp":1688434338000},"page":"1358","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Some Explicit Properties of Frobenius\u2013Euler\u2013Genocchi Polynomials with Applications in Computer Modeling"],"prefix":"10.3390","volume":"15","author":[{"given":"Noor","family":"Alam","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","family":"Khan","sequence":"additional","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"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7958-4226","authenticated-orcid":false,"given":"Can","family":"K\u0131z\u0131late\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zonguldak B\u00fclent Ecevit University, 67100 Zonguldak, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3096-9464","authenticated-orcid":false,"given":"Sofian","family":"Obeidat","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Deanship of Preparatory Year, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cheon Seoung","family":"Ryoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hannam University, Daejeon 34430, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nabawia Shaban","family":"Diab","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Deanship of Preparatory Year, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Alatawi, M.S., and Khan, W.A. 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