{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T16:29:47Z","timestamp":1777566587733,"version":"3.51.4"},"reference-count":21,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T00:00:00Z","timestamp":1744329600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council of the Republic of China","award":["NSTC 113-2115-M-017-004"],"award-info":[{"award-number":["NSTC 113-2115-M-017-004"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Many properties of special numbers, such as sum formulas, symmetric properties, and their relationships with each other, have been studied in the literature with the help of the Binet formula and generating function. In this paper, higher-order generalized Fibonacci hybrid numbers with q-integer components are defined through the utilization of q-integers and higher-order generalized Fibonacci numbers. Several special cases of these newly established hybrid numbers are presented. The article explores the integration of q-calculus and hybrid numbers, resulting in the derivation of a Binet-like formula, novel identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of hybrid numbers with quantum integer coefficients. Furthermore, new identities for these types of hybrids are obtained using two novel special matrices. To substantiate the findings, numerical examples are provided, generated with the assistance of Maple.<\/jats:p>","DOI":"10.3390\/sym17040584","type":"journal-article","created":{"date-parts":[[2025,4,11]],"date-time":"2025-04-11T06:45:31Z","timestamp":1744353931000},"page":"584","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On Higher-Order Generalized Fibonacci Hybrid Numbers with q-Integer Components: New Properties, Recurrence Relations, and Matrix Representations"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7958-4226","authenticated-orcid":false,"given":"Can","family":"K\u0131z\u0131late\u015f","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zonguldak B\u00fclent Ecevit University, Zonguldak 67100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2349-8978","authenticated-orcid":false,"given":"Emrah","family":"Polatl\u0131","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zonguldak B\u00fclent Ecevit University, Zonguldak 67100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0408-8797","authenticated-orcid":false,"given":"Nazl\u0131han","family":"Terzio\u011flu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zonguldak B\u00fclent Ecevit University, Zonguldak 67100, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8996-2270","authenticated-orcid":false,"given":"Wei-Shih","family":"Du","sequence":"additional","affiliation":[{"name":"Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824004, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications, John Wiley & Sons.","DOI":"10.1002\/9781118742297"},{"key":"ref_2","first-page":"414","article-title":"Recurrence relations for the sections of the generating series of the solution to the multidimensional difference equation","volume":"31","author":"Lyapin","year":"2021","journal-title":"Vestn. Udmurtsk. Univ. Mat. Mekh."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"015303","DOI":"10.1088\/1751-8113\/45\/1\/015303","article-title":"Golden quantum oscillator and Binet-Fibonacci calculus","volume":"45","author":"Pashaev","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_4","unstructured":"\u00d6zvatan, M. (2018). Generalized Golden-Fibonacci Calculus and Applications. 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A Note on Generalized k-order F&L Hybrinomials. Axioms, 14.","DOI":"10.3390\/axioms14010041"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/584\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:12:47Z","timestamp":1760029967000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/4\/584"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,11]]},"references-count":21,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["sym17040584"],"URL":"https:\/\/doi.org\/10.3390\/sym17040584","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,11]]}}}