EN
Sum and Weighted Sum Formulas for Fibonacci and Lucas Quaternions
Abstract
Fibonacci and Lucas quaternions are deeply studied in the literature after its definition by Horadam in 1963. Binet formula, generation function, Cassini, Catalan, Honsberger, and other identities for these sequences studied in different quaternion algebra types like split quaternion algebra, dual quaternion algebra, etc. Additionally, some of the generalizations for Fibonacci and Lucas quaternions are studied like k-Fibonacci quaternions, k-Lucas quaternions, Horadam quaternions, and more. However, despite being researched intensively, the summation formulas or weighted summation formulas for these sequences are not examined thoroughly. In this study, after briefly summarizing the literature, we focused on the summation and the weighted summation formulas for the Fibonacci and the Lucas quaternions. In addition, we provided generalized weighted summation formulas for the Fibonacci and Lucas quaternions. Moreover, we calculated generalized weighted summation formulas for the double coefficient Fibonacci and double coefficient Lucas quaternions which have two Fibonacci and Lucas coefficient in every unit of the quaternion, respectively.
Keywords
Ethical Statement
The author declares that he has no conflict of interest.
References
- [1] Bayro-Corrochano E., (2021). A survey on quaternion algebra and geometric algebra applications in engineering and computer science 1995–2020. IEEE Access, 9: 104326–104355.
- [2] Huang C., Li J., Gao G., (2023). Review of quaternion-based color image processing methods. Mathematics, 11: 2056.
- [3] Salamin E., (1979). Application of quaternions to computation with rotations. Tech. rep., Working Paper.
- [4] Horadam A.F., (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70: 289–291.
- [5] Iyer M.R., (1969). Some results on Fibonacci quaternions. The Fibonacci Quarterly, 7: 201–210.
- [6] Halici S., (2012). On Fibonacci quaternions. Adv. Appl. Clifford Algebras, 22: 321–327.
- [7] Polatli E., Kesim S., (2015). On quaternions with generalized Fibonacci and Lucas number components. Advances in Difference Equations, 1: 1–8.
- [8] Ramirez J.L., (2015). Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. Analele stiintifice ale Universitatii Ovidius Constanta. Seria Matematica, 23: 201–212.
Details
Primary Language
English
Subjects
Numerical and Computational Mathematics (Other)
Journal Section
Research Article
Authors
Adnan Karataş
*
0000-0003-3652-5354
Türkiye
Publication Date
December 30, 2024
Submission Date
October 31, 2024
Acceptance Date
December 27, 2024
Published in Issue
Year 1970 Volume: 8 Number: 2
