{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T08:25:22Z","timestamp":1774945522534,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,29]],"date-time":"2024-12-29T00:00:00Z","timestamp":1735430400000},"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 study, we establish some properties of Bronze Fibonacci and Bronze Lucas sequences. Then we find the relationships between the roots of the characteristic equation of these sequences with these sequences. What is interesting here is that even though the roots change, equality is still maintained. Also, we derive the special relations between the terms of these sequences. We give the important relations among these sequences, positive and negative index terms, with the sum of the squares of two consecutive terms being related to these sequences. In addition, we present the application of generalized Bronze Fibonacci sequences to hyperbolic quaternions. For these hyperbolic quaternions, we give the summation formulas, generating functions, etc. Moreover, we obtain the Binet formulas in two different ways. The first is in the known classical way and the second is with the help of the sequence\u2019s generating functions. In addition, we calculate the special identities of these hyperbolic quaternions. Furthermore, we examine the relationships between the hyperbolic Bronze Fibonacci and Bronze Lucas quaternions. Finally, the terms of the generalized Bronze Fibonacci sequences are associated with their hyperbolic quaternion values.<\/jats:p>","DOI":"10.3390\/axioms14010014","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T07:00:37Z","timestamp":1735628437000},"page":"14","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4188-7248","authenticated-orcid":false,"given":"Engin","family":"\u00d6zkan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, Marmara University, \u0130stanbul 34722, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9716-9424","authenticated-orcid":false,"given":"Hakan","family":"Akku\u015f","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Graduate School of Natural and Applied Sciences, Erzincan Binali Y\u0131ld\u0131r\u0131m University, Erzincan 24050, T\u00fcrkiye"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8824-9163","authenticated-orcid":false,"given":"Alkan","family":"\u00d6zkan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Arts and Sciences, I\u011fd\u0131r University, I\u011fd\u0131r 76000, T\u00fcrkiye"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,29]]},"reference":[{"key":"ref_1","first-page":"195","article-title":"The moore-penrose inverse of the rectangular fibonacci matrix and applications to the cryptology","volume":"40","year":"2023","journal-title":"Adv. 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