{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T01:12:56Z","timestamp":1767575576150,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T00:00:00Z","timestamp":1727654400000},"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":["Axioms"],"abstract":"<jats:p>In this paper, by using q-integers and higher-order generalized Fibonacci numbers, we define the higher-order generalized Fibonacci quaternions with q-integer components. We give some special cases of these newly established quaternions. This article examines q-calculus and quaternions together. We obtain a Binet-like formula, some new identities, a generating function, a recurrence relation, an exponential generating function, and sum properties of quaternions with quantum integer coefficients. In addition, we obtain some new identities for these types of quaternions by using three new special matrices.<\/jats:p>","DOI":"10.3390\/axioms13100677","type":"journal-article","created":{"date-parts":[[2024,9,30]],"date-time":"2024-09-30T10:00:47Z","timestamp":1727690447000},"page":"677","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["New Properties and Matrix Representations on Higher-Order Generalized Fibonacci Quaternions with q-Integer Components"],"prefix":"10.3390","volume":"13","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, T\u00fcrkiye"}],"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"}]},{"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, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5956-122X","authenticated-orcid":false,"given":"Ren-Chuen","family":"Chen","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":[[2024,9,30]]},"reference":[{"unstructured":"Hamilton, W.R. (1866). Elements of Quaternions, Longman, Green, & Company.","key":"ref_1"},{"doi-asserted-by":"crossref","unstructured":"Rodman, L. (2014). Topics in Quaternion Linear Algebra, Princeton University Press.","key":"ref_2","DOI":"10.23943\/princeton\/9780691161853.001.0001"},{"doi-asserted-by":"crossref","unstructured":"Vince, J. (2011). Quaternions for Computer Graphics, Springer.","key":"ref_3","DOI":"10.1007\/978-0-85729-760-0"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"109961","DOI":"10.1016\/j.chaos.2020.109961","article-title":"Some results on Horadam quaternions","volume":"138","author":"Tan","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"289","DOI":"10.2307\/2313129","article-title":"Complex Fibonacci numbers and Fibonacci quaternions","volume":"70","author":"Horadam","year":"1963","journal-title":"Am. Math. Mon."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s00006-011-0317-1","article-title":"On Fibonacci quaternions","volume":"22","author":"Halici","year":"2012","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s00006-012-0337-5","article-title":"On complex Fibonacci quaternions","volume":"23","author":"Halici","year":"2013","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_8","first-page":"23","article-title":"Quaternion recurrence relations","volume":"2","author":"Horadam","year":"1993","journal-title":"Ulam Q."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1080\/00150517.1969.12431171","article-title":"A note on Fibonacci quaternions","volume":"7","author":"Iyer","year":"1969","journal-title":"Fibonacci Quart."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1186\/s13662-015-0511-x","article-title":"On quaternions with generalized Fibonacci and Lucas number components","volume":"2015","author":"Polatli","year":"2015","journal-title":"Adv. Differ. Eq."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1007\/s00006-015-0626-x","article-title":"A generalization of Fibonacci and Lucas Quaternions","volume":"26","author":"Polatli","year":"2016","journal-title":"Adv. Appl. Clifford Algebr."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.chaos.2017.03.037","article-title":"On a generalization for quaternion sequences","volume":"98","author":"Halici","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1007\/s41478-020-00295-1","article-title":"On higher order Fibonacci quaternions","volume":"29","author":"Kizilates","year":"2021","journal-title":"J. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"111044","DOI":"10.1016\/j.chaos.2021.111044","article-title":"On higher order Fibonacci hyper complex numbers","volume":"148","author":"Kizilates","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_15","first-page":"263","article-title":"On quaternions with incomplete Fibonacci and Lucas Components","volume":"110","author":"Kizilates","year":"2019","journal-title":"Util. Math."},{"doi-asserted-by":"crossref","unstructured":"Terzio\u011flu, N., K\u0131z\u0131late\u015f, C., and Du, W.-S. (2022). New Properties and Identities for Fibonacci Finite Operator Quaternions. Mathematics, 10.","key":"ref_16","DOI":"10.3390\/math10101719"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.1007\/s41478-024-00730-7","article-title":"On 3-parameter quaternions with higher order generalized Fibonacci numbers components","volume":"32","author":"Kibar","year":"2024","journal-title":"J. Anal."},{"unstructured":"\u00d6zvatan, M. (2018). Generalized golden-Fibonacci Calculus and Applications. [Master\u2019s Thesis, Izmir Institute of Technology].","key":"ref_18"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1142\/S0219887821500754","article-title":"Quantum calculus of Fibonacci divisors and infinite hierarchy of bosonic-fermionic golden quantum oscillators","volume":"18","author":"Pashaev","year":"2021","journal-title":"Internat. J. Geom. Methods Modern Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1017\/S0080456800002751","article-title":"On q-functions and a certain difference operator","volume":"46","author":"Jackson","year":"1909","journal-title":"Earth Environ. Sci. Trans. Roy. Soc. Edin."},{"doi-asserted-by":"crossref","unstructured":"Kac, V., and Cheung, P. (2002). Quantum Calculus Universitext, Springer.","key":"ref_21","DOI":"10.1007\/978-1-4613-0071-7"},{"key":"ref_22","first-page":"1","article-title":"Quaternions: Quantum calculus approach with applications","volume":"46","author":"Akkus","year":"2019","journal-title":"Kuwait J. Sci."},{"key":"ref_23","first-page":"336","article-title":"On quaternions with higher order Jacobsthal numbers components","volume":"36","author":"Uysal","year":"2022","journal-title":"Gazi Univ. J. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2443","DOI":"10.1007\/s41478-023-00579-2","article-title":"On hyper complex numbers with higher order Pell numbers components","volume":"31","year":"2023","journal-title":"J. Anal."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s00006-015-0571-8","article-title":"On Pell Quaternions and Pell-Lucas Quaternions","volume":"26","year":"2016","journal-title":"Adv. Appl. Clifford Algebras"},{"key":"ref_26","first-page":"1","article-title":"A note on Catalan\u2019s identity for the k-Fibonacci Quaternions","volume":"18","author":"Polatli","year":"2015","journal-title":"J. Integer Seq."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/00150517.1969.12431171","article-title":"Some results on Fibonacci quaternions","volume":"7","author":"Iyer","year":"1969","journal-title":"Fibonacci Quart."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/677\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:07:52Z","timestamp":1760112472000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/10\/677"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,30]]},"references-count":27,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["axioms13100677"],"URL":"https:\/\/doi.org\/10.3390\/axioms13100677","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,9,30]]}}}