On Some Properties of Bihyperbolic Numbers of The Lucas Type

Yıl 2023, Cilt: 6 Sayı: 4, 226 - 239, 25.12.2023

Fügen Torunbalcı Aydın

https://doi.org/10.33434/cams.1372245

Öz

To date, many authors in the literature have worked on special arrays in various computational systems. In this article, Lucas type bihyperbolic numbers were defined and their algebraic properties were examined. Bihyperbolic Lucas numbers were studied by Azak in 2021. Therefore, we only examined bihyperbolic Jacobsthal-Lucas and Pell-Lucas numbers. We also gave properties of bihyperbolic Jacobstal-Lucas and bihyperbolic Pell-Lucas numbers such as recursion relation, derivation function, Binet formula, D'Ocagne identity, Cassini identity and Catalan identity.

Anahtar Kelimeler

Bihyperbolic number, Bihyperbolic Jacobsthal-Lucas number, Bihyperbolic Pell-Lucas number, Jacobsthal-Lucas number, Lucas number, Pell-Lucas number

Kaynakça

• [1] W. R. Hamilton, Lectures on Quaternions, Hodges and Smith. Dublin, 1853.
• [2] J. Cockle, On certain functions resembling quaternions and on a new imaginary in algebra., The London, Edinburg and Dublin Philosophical Mag. J. Sci., 33 (1848), 435-439.
• [3] C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici., Math. Ann., 40 (1892), 413-467.
• [4] F. Catoni, D. Boccaletti, C. V. Cannata, Catoni, E. Nichelatti, F. Zampetti, The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers, Basel, Boston, Berlin: Birkhauser Verlag, 2008.
• [5] G. B. Price, An Introduction to Multicomplex Spaces and Functions, M. Dekker New York, 1991.
• [6] A. A. Pogorui, R. M. Rodrigez-Dagnino, R. D. Rodrigez-Said, On the set of zeros of bihyperbolic polynomials., Complex Var. Elliptic Equ., 53, (2008), 685-690.
• [7] S. Olariu, Complex Number in n-dimensions, Nerth-Holland Mathematics Studies, 190, Amsterdam, Boston: Elsevier, 51-148, 2002.
• [8] M. Bilgin, S. Ersoy, Algebraic properties of bihyperbolic numbers., Adv. Appl. Clifford Algebr., 30 (2020), 1-17.
• [9] N. Gürses, G. Y. Sentürk, S. Yüce, A study on dual-generalized complex and hyperbolic-generalized complex numbers., Gazi Univ. J. Sci., 34 (2021), 180-194.
• [10] D. Brod, A. Syznal-Liana, I. Wloch, On some combinatorial properties of bihyperbolic numbers of the Fibonacci type., Math. Meth. App. Sci., 44 (2021), 4607-4615.
• [11] D. Brod, A. Syznal-Liana, I. Wloch, On a new generalization of bihyperbolic Pell numbers., Ann. Alexandru Ioan Cuza Univ. Math., 67(2) (2021).
• [12] A. Z. Azak, Some new identities with respect to bihyperbolic Fibonacci and Lucas numbers., Int. J. Sci.: Basic and App. Res., 60 (2021), 14-37.
• [13] A. Szynal-Liana, I. Włoch, On Jacobsthal and Jacobsthal-Lucas hybrid numbers., Ann. Math. Sil., 33 (2019), 276-283.
• [14] A. F. Horadam, Jacobsthal representation numbers., Fibonacci Quart., 34 (1996), 40-54.
• [15] A. F. Horadam, Jacobsthal representation polynomials., Fibonacci Quart., 35 (1997), 137-148.
• [16] F. T. Aydın, On generalizations of the Jacobsthal sequence., Notes Number Theory Discrete Math., 24(1) (2018), 120-135.
• [17] S. Uygun, A new generalization for Jacobsthal and Jacobsthal-Lucas sequences., Asian J. Math., 2(1) (2018), 14-21.
• [18] A. Al-Kateeb, A generalization of Jacobsthal and Jacobsthal-Lucas numbers., (2019), arXiv preprint:1911.11515.
• [19] D. Brod, A. Michalski, On generalized Jacobsthal and Jacobsthal-Lucas numbers., Ann. Math. Sil., 36(2) (2022), 115-128.
• [20] A.F. Horadam, Pell identities., Fibonacci Quart., 9(3) (1971), 245-252.
• [21] A. F. Horadam, J. Mahon, Pell and Pell-Lucas polynomials., Fibonacci Quart., 23(1) (1985), 7-20.
• [22] S. F. Santana, et al, Some properties of sums involving Pell numbers, Missouri J. Math. Sci. Uni. Central Missouri, Department of Mathematics and Computer Science, 18(1) (2006), 33-40.
• [23] A. Szynal-Liana, I. Włoch, On certain bihypernomials related to Pell and Pell-Lucas numbers., Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 71(2) (2022), 422-433.
• [24] H. Gökbaş, Gaussian-bihyperbolic numbers containing Pell and Pell-Lucas numbers., J. Adv. Res. Nat. App. Sci., Çanakkale Onsekiz Mart Univ. 9(1) (2023), 183-189.
• [25] F. T. Aydın, On bicomplex Pell and Pell-Lucas numbers., Comm. Adv. Math. Sci., 1(2) (2018), 142-155.