On the structure and generalization of bihyperbolic Leonardo sequences

Hasan Gökbaş; Anetta Szynal-Liana

doi:10.5269/bspm.77488

Hasan Gökbaş Gökbaş
Anetta Szynal-Liana Rzeszow University of Technology Faculty of Mathematics and Applied Physics

Abstract

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.

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References

H. Akku¸s, E. Ozkan, Generalization of the k-Leonardo sequence and their hyperbolic quaternions, Math. Montisnigri, 14–31, (2024). https://doi.org/10.20948/mathmontis-2024-60-2

Y. Alp, E.G. Kocer, Hybrid Leonardo numbers, Chaos Solitons Fractals, 150, 111128, (2021). https://doi.org/10.1016/j.chaos.2021.111128

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

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

P. M. M. C. Catarino, A. Borges, On Leonardo Numbers, Acta Math. Univ. Comenianae, 1, 75–86, (2020).

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

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).

N. Kara, F. Yılmaz, On hybrid numbers with Gaussian Leonardo coefficients, Mathematics, 11(6), 1551, (2023).

A. Karatas, Dual Leonardo numbers, AIMS Mathematics, 8(12), 30527–30536, (2023). https://doi.org/10.3934/math.20231560

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

E. Ozkan, H. Akkus, Generalized Bronze Leonardo sequence, Notes Number Theory Discrete Math, 30(4), 811-824, (2024).

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).

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).

The On-Line Encyclopedia of Integer Sequences, A001595.

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