Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
quaternion; elliptic quaternion; Fibonacci number; Leonardo number
quaternion; elliptic quaternion; Fibonacci number; Leonardo number
Summary:
The motion of a point on an ellipsoid can be interpreted using the generation of elliptical rotations. An elliptical rotation can be defined using elliptic quaternions. The Fibonacci sequence, defined by the second-order linear relation, is a famous sequence used in mathematics, physics, computer science, technical analysis and others. In this paper, we introduce and study generalized Fibonacci-Leonardo elliptic quaternions, which generalize the Fibonacci elliptic quaternions and Leonardo elliptic quaternions, simultaneously. We give the ordinary generating functions, the Binet-type formulas, general bilinear index-reduction formulas, and the sum of the finite, finite odd, and finite even terms of these quaternions.
References:
[1] Alp, Y.: On the generalized Francois numbers. Turk. J. Math. Comput. Sci. 16 (2024), 346-353. DOI 10.47000/tjmcs.1464650
[2] Bednarz, U., Wołowiec-Musiał, M.: Generalized Fibonacci-Leonardo numbers. J. Difference Equ. Appl. 30 (2024), 111-121. DOI 10.1080/10236198.2023.2265509 | MR 4712919 | Zbl 1543.11020
[3] Catarino, P. M. M. C., Borges, A.: On Leonardo numbers. Acta Math. Univ. Comen., New Ser. 89 (2020), 75-86. MR 4053337 | Zbl 1481.11016
[4] Dişkaya, O., Menken, H., Catarino, P. M. M. C.: On the hyperbolic Leonardo and hyperbolic Francois quaternions. J. New Theory 42 (2023), 74-85. DOI 10.53570/jnt.1199465
[5] Gökbaş, H.: A new generalization of Leonardo sequences: Biperiodic Leonardo sequence. J. Math. 2025 (2025), Article ID 3399017, 9 pages. DOI 10.1155/jom/3399017 | MR 4882547
[6] Hamilton, W. R.: On a new species of imaginary quantities, connected with a theory of quaternions. Math. Proc. R. Ir. Acad. 2 (1844), 424-434.
[7] Koshy, T.: Fibonacci and Lucas Numbers with Applications. Vol. I. John Wiley & Sons, New York (2001). DOI 10.1002/9781118033067 | MR 1855020 | Zbl 0984.11010
[8] Kuhapatanakul, K., Chobsorn, J.: On the generalized Leonardo numbers. Integer 22 (2022), Article ID A48, 7 pages. MR 4430899 | Zbl 1493.11034
[9] "Ozdemir, M.: An alternative approach to elliptical motion. Adv. Appl. Clifford Algebr. 26 (2016), 279-304. DOI 10.1007/s00006-015-0592-3 | MR 3460002 | Zbl 1380.15019
[10] Prasad, K., Mohanty, R., Kumari, M., Mahato, H.: Some new families of generalized $k$-Leonardo and Gaussian Leonardo numbers. Commun. Comb. Optim. 9 (2024), 539-553. DOI 10.22049/cco.2023.28236.1485 | MR 4758515 | Zbl 1560.11024
[11] Seçgin, F.: On Leonardo, Leonardo-Lucas and modified Leonardo elliptic quaternions and their matrix representations. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 79 (2025), 85-103. DOI 10.17951/a.2025.79.1.85-103 | MR 4964598 | Zbl 1575.11042
[12] Soykan, Y.: Generalized Leonardo numbers. J. Progres. Res. Math. 18 (2021), 58-84.
[13] Tan, E., Yaman, T., Gök, I.: On Fibonacci and Lucas elliptic quaternions. J. Sci. Arts 25 (2025), 31-44. DOI 10.46939/j.sci.arts-25.1-a04
[14] Vajda, S.: Fibonacci & Lucas Numbers, and the Golden Section: Theory and Applications. John Wiley & Sons, New York (1989). MR 1015938 | Zbl 0695.10001
Search
Partner of
