A note on hyperbolic (p,q)-Fibonacci quaternions

Abstract

In this paper, we introduce a new quaternion sequence called  hyperbolic (p,q)-Fibonacci  quaternions.  This new quaternion sequence includes hyperbolic Fibonacci, hyperbolic k-Fibonacci, hyperbolic  Pell, hyperbolic k-Pell, hyperbolic  Jacobsthal, hyperbolic k-Jacobsthal quaternions. We give generating function and Binet formula for these quaternions. We also obtain   some  identities such as d'Ocagne's, Catalan's and Cassini's identities involving  hyperbolic (p,q)-Fibonacci  quaternions.

Keywords

• Fibonacci numbers,(p;q)-Fibonacci numbers,(p;q)-Fibonacci quaternions,hyperbolic quaternions,hyperbolic (p;q)-Fibonacci quaternions

References

1. Adler, S. L., Quaternionic quantum mechanics and quantum fields, 88. Oxford Univ. Press on Demand, 1995.
2. Akyigit, M., Kosal, H. H. and Tosun, M., Split Fibonacci quaternions, Adv. Appl. Clifford Algebras 23 (2013), 535-545.
3. Aydin, F. T., The k-Fibonacci dual quaternions, Int. J. Mathematical Analysis 12 (2018), 363-373.
4. Aydin, F. T., Bicomplex Fibonacci quaternions, Chaos Solitons Fractals 106 (2018), 147-153.
5. Aydin, F. T., Hyperbolic k-Fibonacci quaternions, arXiv:1812.00781v1, 2018.
6. Bilgici, G., Tokeser, U. and Unal, Z., k- Fibonacci and k-Lucas generalized quaternions, Konuralp J. Math. 5 (2017), 102-113.
7. Catarino, P., On some identities and generating functions for k-Pell numbers, Int. J. Mathematical Analysis 7 (2013), 1877-1884.
8. Ell, T. A., Bihan, N. L. and Sangwine, S. J., Quaternion Fourier transforms for signal and image processing, John Wiley and Sons, 2014.

9. Falcon, S. and Plaza, A., On the Fibonacci k-numbers, Chaos Solitons Fractals 32 (2007), 1615-1624.
10. Finkelstein, D., Jauch, J. M., Schiminovich, S. and Speiser, D., Foundations of quaternion quantum mechanics, J. Math. Phys. 3 (1962), 207-220.
11. Girard, P. R., The quaternion group and modern physics, Eur. J. Phys. 5 (1984), 25-32.
12. Halici, S., On Fibonacci quaternions, Adv. Appl. Clifford Algebras 22 (2012), 321-327.
13. Halici, S., On complex Fibonacci quaternions, Adv. Appl. Clifford Algebras 23 (2013), 105-112.
14. Hamilton, W. R., Lectures on quaternions, Hodges and Smith, Dublin, 1853.
15. Horadam, A. F., A generalized Fibonacci sequence, The American Math. Monthly 68 (1961), 455-459.
16. Horadam, A. F., Complex Fibonacci numbers and Fibonacci quaternions, The American Math. Monthly 70 (1963), 289-291.
17. Horadam, A. F., Basic properties of a certain generalized sequence of numbers, The Fibonacci Quart. 3 (1965), 161-176.
18. Horadam, A. F., Pell identities, The Fibonacci Quart. 9 (1971), 245-252.
19. Horadam, A. F., Jacobsthal representation numbers, The Fibonacci Quart. 34 (1996), 40-54.
20. Ipek, A., On (p,q)-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums, Adv. Appl. Clifford Algebras 27 (2017), 1343-1351.
21. Iyer, M. R., A note on Fibonacci quaternions, The Fibonacci Quart. 7 (1969), 225-229.
22. Jhala, D., Sisodiya, K. and Rathore, G. P. S., On some identities for k-Jacobsthal numbers, Int. J. Mathematical Analysis 7 (2013), 551-556.
23. Kosal, I. A., A note on hyperbolic quaternions, Universal Journal of Mathematics and Applications 1 (2018), 155-159.
24. Koshy, T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, New York, 2001.
25. Mac-Farlane, A., Hyperbolic quaternions, Proc. Roy. Soc. Edinburg 23 (1902), 169-180.
26. Patel, B. K. and Ray, P. K., On the properties of (p,q)-Fibonacci and (p,q)-Lucas quaternions, Mathematical Reports 21 (2019), 15-25.
27. Polatli, E., Kizilates, C. and Kesim, S., On split k-Fibonacci and k-Lucas quaternions, Adv. Appl. Clifford Algebras 26 (2016), 353-362.
28. Ramirez, J. L., Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions, An. St. Univ. Ovidius Constanta 23 (2015), 201-212.
29. Ribenboim, P., My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, New York, 2000.
30. Senturk, T. D., Dasdemir, A., Bilgici, G. and Unal, Z., On unrestricted Horadam generalized quaternions, Utilitas Mathematica 110 (2019), 89-98.
31. Tan, E., Yilmaz, S. and Sahin, M., On a new generalization of Fibonacci quaternions, Chaos Solitons Fractals 82 (2016), 1-4.
32. Ward, J. P., Quaternions and Cayley Numbers: Algebra and Applications, Kluwer, London, 1997.