On the structure and generalization of bihyperbolic Leonardo sequences
DOI:
https://doi.org/10.5269/bspm.77488Abstract
In this paper, we give some properties of the bihyperbolic Leonardo numbers, among others the Binet formula, generating function formula and the general bilinear index-reduction formula which implies d'Ocagne, Vajda, Halton, Catalan, and Cassini identities. We also give the matrix representation and some sum formulas of the bihyperbolic Leonardo numbers. Moreover, we present a one-parameter generalization of the bihyperbolic Leonardo numbers and their properties.
References
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2. Y. Alp, E.G. Kocer, Hybrid Leonardo numbers, Chaos Solitons Fractals, 150, 111128, (2021). https://doi.org/10.1016/j.chaos.2021.111128
3. U. Bednarz, M. Wolowiec-Musial, Generalized Fibonacci-Leonardo numbers, J. Difference Equ. Appl., 30:1, 111–121, (2024). http://doi.org/10.1080/10236198.2023.2265509
4. M. Bilgin, S. Ersoy, Algebraic properties of bihyperbolic numbers, Adv. Appl. Clifford Algebr., 30:13, (2020). https://doi.org/10.1007/s00006-019-1036-2
5. P. M. M. C. Catarino, A. Borges, On Leonardo Numbers, Acta Math. Univ. Comenianae, 1, 75–86, (2020).
6. C. M. Dikmen, A Study on Dual Hyperbolic Generalised Leonardo Numbers, Asian Research Journal of Mathematics, 21(6), 143–165, (2025). https://doi.org/10.9734/arjom/2025/v21i6950
7. O. Diskaya, H. Menken, P. M. M. C. Catarino, On the hyperbolic Leonardo and hyperbolic Francois quaternions, Journal of New Theory, (42), 74-85, (2023).
8. N. Kara, F. Yılmaz, On hybrid numbers with Gaussian Leonardo coefficients, Mathematics, 11(6), 1551, (2023).
9. A. Karatas, Dual Leonardo numbers, AIMS Mathematics, 8(12), 30527–30536, (2023). https://doi.org/10.3934/math.20231560
10. A. Karatas, On complex Leonardo numbers, Notes on Number Theory and Discrete Mathematics, 28(3), 458–465, (2022). https://doi.org/10.7546/nntdm.2022.28.3.458-465
11. E. Ozkan, H. Akkus, Generalized Bronze Leonardo sequence, Notes Number Theory Discrete Math, 30(4), 811-824, (2024).
12. E. V. P. Spreafico, P. M. M. C. Catarino, A note on hybrid hyper Leonardo numbers, Turkish Journal of Mathematics, 48(3), 557-566, (2024).
13. E. Tan, D. Savin, S. Yılmaz, A new class of Leonardo hybrid numbers and some remarks on Leonardo quaternions over finite fields, Mathematics, 11(22), 4701, (2023).
14. The On-Line Encyclopedia of Integer Sequences, A001595.
15. M. Turan, S. O. Karakus, S. K. Nurkan, A new perspective on bicomplex numbers with Leonardo number components, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 72(2), 340–351, (2023). https://doi.org/10.31801/cfsuasmas.1181930
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2025-09-02
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