{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T21:38:07Z","timestamp":1765057087103,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,5,24]],"date-time":"2024-05-24T00:00:00Z","timestamp":1716508800000},"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>This paper presents an overview of cosine and sine Apostol-type Frobenius\u2013Euler\u2013Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius\u2013Euler\u2013Fibonacci polynomials of order \u03b1 and several other polynomial sequences, such as the Apostol-type Bernoulli\u2013Fibonacci polynomials, the Apostol-type Euler\u2013Fibonacci polynomials, the Apostol-type Genocchi\u2013Fibonacci polynomials, and the Stirling\u2013Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program.<\/jats:p>","DOI":"10.3390\/axioms13060348","type":"journal-article","created":{"date-parts":[[2024,5,24]],"date-time":"2024-05-24T05:36:06Z","timestamp":1716528966000},"page":"348","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Certain Properties of Parametric Kinds of Apostol-Type Frobenius\u2013Euler\u2013Fibonacci Polynomials"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7859-5409","authenticated-orcid":false,"given":"Hao","family":"Guan","sequence":"first","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"},{"name":"School of Computer Science of Information Technology, Qiannan Normal University for Nationalities, Duyun 558000, China"}]},{"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"}]},{"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, Zonguldak B\u00fclent Ecevit University, Zonguldak 67100, Turkey"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4647-1380","authenticated-orcid":false,"given":"Cheon Seoung","family":"Ryoo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hannam University, Daejeon 34430, Republic of Korea"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Alam, N., Khan, W.A., and Ryoo, C.S. (2022). A note on Bell-based Apostol-type Frobenius-Euler polynomials of complex variable with its certain applications. 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