Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions
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 mainta...
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2024-12-01
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| author | Engin Özkan Hakan Akkuş Alkan Özkan |
| author_facet | Engin Özkan Hakan Akkuş Alkan Özkan |
| author_sort | Engin Özkan |
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| description | 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’s 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. |
| format | Article |
| id | doaj-art-f913e9a5b6484bb2b9f4a6003bc3de2f |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-f913e9a5b6484bb2b9f4a6003bc3de2f2025-01-24T13:22:09ZengMDPI AGAxioms2075-16802024-12-011411410.3390/axioms14010014Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic QuaternionsEngin Özkan0Hakan Akkuş1Alkan Özkan2Department of Mathematics, Faculty of Sciences, Marmara University, İstanbul 34722, TürkiyeDepartment of Mathematics, Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University, Erzincan 24050, TürkiyeDepartment of Mathematics, Faculty of Arts and Sciences, Iğdır University, Iğdır 76000, TürkiyeIn 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’s 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.https://www.mdpi.com/2075-1680/14/1/14Bronze Fibonacci numberBronze Lucas numberquaternionsgenerating functionCatalan identity |
| spellingShingle | Engin Özkan Hakan Akkuş Alkan Özkan Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions Axioms Bronze Fibonacci number Bronze Lucas number quaternions generating function Catalan identity |
| title | Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions |
| title_full | Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions |
| title_fullStr | Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions |
| title_full_unstemmed | Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions |
| title_short | Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions |
| title_sort | properties of generalized bronze fibonacci sequences and their hyperbolic quaternions |
| topic | Bronze Fibonacci number Bronze Lucas number quaternions generating function Catalan identity |
| url | https://www.mdpi.com/2075-1680/14/1/14 |
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