On Gaussian Quadra Fibona-Pell Sequence and A Quaternion Sequence Formed by the Terms of This Sequence

Abstract

In this study, the Gaussian quadra Fibona-Pell sequence is proposed and examined. The quadra Fibona-Pell sequence is first extended to define the Gaussian quadra Fibona-Pell sequence. Then the generating function, Binet-like formula, and some identities are represented. In addition, some formulas related to the Gaussian quadra Fibona-Pell sequence and some matrices containing terms of the sequence are studied. Finally we define a quaternion sequence formed by the terms of Gaussian quadra Fibona-Pell sequence.

Keywords

• Fourth-order recurrence relation
• Gaussian numbers
• Quaternion.

References

1. [1] T. Koshy, “Fibonacci and Lucas numbers with applications, ” John Wiley and Sons, vol. 1, 2018.
2. [2] C. Gauss, “Theoria residuorum biquadraticorum,” Commentatio prima, Typis Dieterichchianis, 1832. [3] A. F. Horadam, “Complex Fibonacci numbers and Fibonacci quaternions,” American Mathematical Monthly, vol. 70, pp. 289-291, 1963.
3. [4] J. H. Jordan, “Gaussian Fibonacci and Lucas numbers,” The Fibonacci Quarterly, vol. 3, pp. 315-318, 1965.
4. [5] S. Halıcı and S. Öz, “On some Gaussian Pell and Pell–Lucas numbers,” Ordu University Journal of Science and Technology, vol. 6, no.1, pp. 8-18, 2016.
5. [6] G. Berzsenyi, “Gaussian Fibonacci numbers,” The Fibonacci Quarterly, vol. 15, no. 3, pp. 233-236, 1997.
6. [7] C. J. Harman, “Complex Fibonacci numbers,” The Fibonacci Quarterly, vol. 19, no.1, 82-86, 1981.
7. [8] S. Pethe and A. F. Horadam, “Generalised Gaussian Fibonacci numbers,” Bulletin of the Australian Mathematical Society, vol. 33, no. 1, pp. 37-48, 1986.
8. [9] D. Taşcı, “On Gaussian Mersenne numbers,” Journal of Science and Arts, vol. 57, no. 4, pp. 1021-1028, 2021.

9. [10] D. Taşcı, “Gaussian Padovan and Gaussian Pell-Padovan Sequences,” Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, vol. 67, no. 2, pp. 82-88, 2018.
10. [11] Y. Soykan, “Linear summing formulas of generalised pentanacci and Gaussian generalised pentanacci numbers,” Journal of Advances in Mathematics and Computer Science, vol. 33, no. 3, pp. 1-14, 2022.
11. [12] N. Saba and A. Boussayoud, “Gaussian (p,q)-Jacobsthal and Gaussian (p,q)-Jacobsthal Lucas numbers and their some interesting properties,” Journal of Mathematical Physics, vol. 63, no. 11, pp. 1-17, 2022.
12. [13] K. Prasad, R. Mohanty, M. Kumari, and H. Mahato, “Some new families of generalized k-Leonardo and Gaussian Leonardo numbers,” Communication in Combinatorics and Optimization, vol. 3, no. 9, pp. 539-553, 2024.
13. [14] A. Özkoç, “Some algebraic identities on quadra Fibona- Pell integer sequence,” Advances in Difference Equation, vol. 2015, no. 148, pp. 1-10, 2015.
14. [15] A. Özkoç Öztürk and E. Gündüz, “Binomial transform for Quadra Fibona-Pell sequence and Quadra Fibona-Pell quaternion,” Universal Journal of Mathematics and Applications, vol. 5, no. 4, pp. 145-155, 2022.
15. [16] S. Halıcı, “On Fibonacci quaternions,” Advances in Applied Clifford Algebras, vol. 22, pp. 321-327, 2012.
16. [17] E. Polatlı, “A generalization of Fibonacci and Lucas quaternions,” Advances in Applied Clifford Algebras, vol. 26, no. 2, pp. 719-730, 2016.
17. [18] S. Halıcı and A. Karataş, “On a generalization for Fibonacci quaternions,” Chaos, Solitons and Fractals, vol. 98, pp. 178-182, 2017.
18. [19] E. Ozkan and M. Uysal, “On quaternions with higher order Jacobsthal numbers components,” Gazi University Journal of Science, vol. 36, no. 1, pp. 336-347, 2023.
19. [20] A. Özkoç Öztürk and F. Kaplan, “Some properties of bivariate Fibonacci and Lucas quaternion polynomials,” Facta Universitatis, Series: Mathematics and Informatics, vol. 35, no. 1, pp. 073-087, 2020.
20. [21] E. Polatlı, “On Certain Properties of Quadrapell Quaternions,” Karaelmas Fen ve Mühendislik Dergisi, vol. 8, no. 1, pp. 305-308, 2018.
21. [22] C. Kızılateş and E. Polatlı, “New families of Fibonacci and Lucas octonions with q-integer components,” Indian Journal of Pure and Applied Mathematics, vol. 52, no. 1, pp. 231-240, 2021.
22. [23] F. Kaplan and A. Özkoç Öztürk, “On the binomial transforms of the Horadam quaternion sequence,” Mathematical Methods in the Applied Sciences , vol. 45, no. 8, pp. 12009-12022, 2022.
23. [24] S. Halıcı and G. Cerda Morales, “On quaternion-Gaussian Fibonacci numbers and their properties,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 29, no. 1, pp. 71-81, 2021.
24. [25] S. Halıcı, “On quaternion-Gaussian Lucas numbers,” Mathematical Methods in the Applied Sciences, vol. 44, no. 9, pp. 7601-7606, 2021.
25. [26] H. Arslan, “Gaussian Pell and Gaussian Pell-Lucas quaternions,” Filomat, vol. 35, no. 5, pp. 1609-1617, 2021.
26. [27] A. Z. Azak, “Pauli Gaussian Fibonacci and Pauli Gaussian Lucas quaternions,” Mathematics, vol. 10, no. 24, pp. 4655, 2022.
27. [28] A. Ertaş and F. Yılmaz, “On quaternions with Gaussian Oresme coefficients,” Turkish Journal of Mathematics and Computer Science, vol. 15, no. 1, pp. 192-202, 2023.
28. [29] A. Sloin and A. Wiesel, “Proper quaternion Gaussian graphical models,” IEEE Transactions on Signal Processing, vol. 62, no. 20, pp. 5487-5496, 2014.