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
Fibonacci Numbers Lucas Numbers Sum Formulas Weighted Sum Formulas Double Coefficient Fibonacci Numbers
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.
- [9] Akyigit M., Hüda Kösal H., Tosun M., (2014). Fibonacci generalized quaternions. Advances in Applied Clifford Algebras, 24: 631–641.
- [10] Bitim B.D., (2019). Some identities of Fibonacci and Lucas quaternions by quaternion matrices. Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 7: 606–615.
- [11] Cerda-Morales G., (2017). Some properties of horadam quaternions. arXiv preprint, arXiv:1707.05918.
- [12] Flaut C., Savin D., (2015). Quaternion algebras and generalized Fibonacci- Lucas quaternions. Advances in Applied Clifford Algebras, 25: 853–862.
- [13] Halici S., Karataş A., (2017). Some matrix representations of Fibonacci quaternions and octonions. Advances in Applied Clifford Algebras, 27: 1233–1242.
- [14] Irmak N., (2020). More identities for fibonacci and lucas quaternions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69: 369–375.
- [15] Köme S., Köme C., Yazlık Y., (2019). Modified generalized Fibonacci and Lucas quaternions. Journal of Science and Arts, 19: 49-60.
- [16] Patel B.K., Ray P.K., (2019). On the properties of (p, q)-Fibonacci and (p, q)-Lucas quaternions. Mathematical Reports, 21: 15–25.
- [17] Polatlı E., (2016). A generalization of Fibonacci and Lucas quaternions. Advances in Applied Clifford Algebras, 26: 719–730.
- [18] Swamy M., (1973). On generalized Fibonacci quaternions. The Fibonacci Quarterly, 11: 547–550.
- [19] Tasci D., Buyukkose S., Kizilirmak G.O., Sevgi E., (2020). Properties of Lucas-sum graph. Journal of Science and Arts, 20: 313–316.
- [20] Kilic E., Stakhov A., (2009). On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices. Chaos, Solitons & Fractals, 40: 2210–2221.
- [21] Koshy T., (2019), Fibonacci and Lucas Numbers with Applications. John Wiley & Sons.
Abstract
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.
- [9] Akyigit M., Hüda Kösal H., Tosun M., (2014). Fibonacci generalized quaternions. Advances in Applied Clifford Algebras, 24: 631–641.
- [10] Bitim B.D., (2019). Some identities of Fibonacci and Lucas quaternions by quaternion matrices. Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 7: 606–615.
- [11] Cerda-Morales G., (2017). Some properties of horadam quaternions. arXiv preprint, arXiv:1707.05918.
- [12] Flaut C., Savin D., (2015). Quaternion algebras and generalized Fibonacci- Lucas quaternions. Advances in Applied Clifford Algebras, 25: 853–862.
- [13] Halici S., Karataş A., (2017). Some matrix representations of Fibonacci quaternions and octonions. Advances in Applied Clifford Algebras, 27: 1233–1242.
- [14] Irmak N., (2020). More identities for fibonacci and lucas quaternions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69: 369–375.
- [15] Köme S., Köme C., Yazlık Y., (2019). Modified generalized Fibonacci and Lucas quaternions. Journal of Science and Arts, 19: 49-60.
- [16] Patel B.K., Ray P.K., (2019). On the properties of (p, q)-Fibonacci and (p, q)-Lucas quaternions. Mathematical Reports, 21: 15–25.
- [17] Polatlı E., (2016). A generalization of Fibonacci and Lucas quaternions. Advances in Applied Clifford Algebras, 26: 719–730.
- [18] Swamy M., (1973). On generalized Fibonacci quaternions. The Fibonacci Quarterly, 11: 547–550.
- [19] Tasci D., Buyukkose S., Kizilirmak G.O., Sevgi E., (2020). Properties of Lucas-sum graph. Journal of Science and Arts, 20: 313–316.
- [20] Kilic E., Stakhov A., (2009). On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices. Chaos, Solitons & Fractals, 40: 2210–2221.
- [21] Koshy T., (2019), Fibonacci and Lucas Numbers with Applications. John Wiley & Sons.
Details
| Primary Language | English |
|---|---|
| Subjects | Numerical and Computational Mathematics (Other) |
| Journal Section | Research Article |
| Authors | |
| Publication Date | December 30, 2024 |
| Submission Date | October 31, 2024 |
| Acceptance Date | December 27, 2024 |
| Published in Issue | Year 2024 |