A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912975106867200 |
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| author | Mantovanelli, Marco |
| author_facet | Mantovanelli, Marco |
| contents | We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19343 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity Mantovanelli, Marco Combinatorics Number Theory 15A24, 11B39, 05A19, 15A15 We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers. |
| title | A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity |
| topic | Combinatorics Number Theory 15A24, 11B39, 05A19, 15A15 |
| url | https://arxiv.org/abs/2603.19343 |