Fundamental properties of extended Horadam numbers

Gülsüm Yeliz Şentürk, Nurten Gürses and Salim Yüce
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 219–235
DOI: 10.7546/nntdm.2021.27.4.219-235
Full paper (PDF, 302 Kb)

Details

Authors and affiliations

Gülsüm Yeliz Şentürk
Faculty of Engineering and Architecture, Istanbul Gelisim University,
Department of Computer Engineering, 34310, Istanbul, Turkey

Nurten Gürses
Faculty of Arts and Sciences, Yildiz Technical University,
Department of Mathematics, 34220, Istanbul, Turkey

Salim Yüce
Faculty of Arts and Sciences, Yildiz Technical University,
Department of Mathematics, 34220, Istanbul, Turkey

Abstract

In this study, we have defined Fibonacci quaternion matrix and investigated its powers. We have also derived some important and useful identities such as Cassini’s identity using this new matrix.

Keywords

• Horadam number
• Dual-generalized complex number
• Hyperbolic-generalized complex number
• Binet’s formula
• D’Ocagne’s identity
• Catalan’s identity
• Cassini’s identity

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 11K31

References

1. Akar, M., Yüce, S., & Sahin, S. (2018). On the dual hyperbolic numbers and the complex hyperbolic numbers. Journal of Computer Science & Computational Mathematics, 8(1), 1–6.
2. Azak, A. Z. (2021). Some new identities with respect to Bihyperbolic Fibonacci and Lucas numbers. International Journal of Sciences: Basic and Applied Sciences, 60, 14–37.
3. Belbachir, H., & Belkhir, A. (2018). On some generalizations of Horadam’s numbers. Filomat, 32(14), 5037–5052.
4. Brod, D., Szynal-Liana, A., & Włoch, I. (2021). On some combinatorial properties of bihyperbolic numbers of the Fibonacci type. Mathematical Methods in the Applied Sciences, 44, 4607–4615.
5. Catoni, F., Boccaletti, D., Cannata, R., Catoni, V., Nichelatti, E., & Zampetti, P. (2008). The Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers. Birkhauser Verlag, Basel, Boston, Berlin.
6. Cheng, H. H., & Thompson, S. (1999). Singularity analysis of spatial mechanisms using dual polynomials and complex dual numbers. Journal of Mechanical Design, 121(2), 200–205.
7. Cihan, A., Azak, A. Z., Gungör, M. A., & Tosun, M. (2019). A study on dual  hyperbolic Fibonacci and Lucas numbers. Analele Universitatii “Ovidius” Constanta-Seria Matematica, 27(1), 35–48.
8. Cockle, J. (1849). On a new imaginary in algebra. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34(226), 37–47.
9. Cohen, A., & Shoham, M. (2018). Principle of transference–An extension to hyper-dual numbers. Mechanism and Machine Theory, 125, 101–110.
10. Fike, J. A., & Alonso, J. J. (2012). Automatic differentiation through the use of hyper-dual numbers for second derivatives. In: Forth S., Hovland P., Phipps E., Utke J., Walther A. (eds) Recent Advances in Algorithmic Differentiation. Lecture Notes in Computational Science and Engineering, Springer, Berlin, Heidelberg, 87, 163–173.
11. Gungör, M. A., & Azak, A. Z. (2017). Investigation of dual-complex Fibonacci,  dual-complex Lucas numbers and their properties. Advances in Applied Clifford Algebras, 27(4), 3083–3096.
12. Gürses, N., Şentürk, G. Y., & Yüce, S. (2021). A study on dual-generalized complex and hyperbolic-generalized complex numbers. Gazi University Journal of Science, 34(1), 180–194.
13. Gürses, N., Şentürk, G. Y., & Yüce, S. (2021). A comprehensive survey of dual-generalized complex Fibonacci and Lucas numbers. Sigma Journal of Engineering and Natural Sciences, (in press).
14. Halıcı, S. (2019). On Bicomplex Fibonacci Numbers and Their Generalization, Models and Theories in Social Systems. Studies in Systems, Decision and Control, 179, Springer, Cham.
15. Harkin, A. A., & Harkin, J. B. (2004). Geometry of generalized complex numbers. Mathematics magazine, 77(2), 118–129.
16. Haukkanen, P. (2002). A note on Horadam’s sequence. The Fibonacci Quaterly, 40(4), 358–361.
17. Horadam, A. F. (1961). A generalized Fibonacci sequence. The American Mathematical Monthly, 68(5), 455–459.
18. Horadam, A. F. (1965). Generating functions for powers of a certain generalised sequence of numbers. Duke Mathematical Journal, 32(3), 437–446.
19. Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The Fibonacci Quarterly, 3, 161–176.
20. Horadam, A. F. (1967). Special properties of the sequence Wn(a, b; p, q). The Fibonacci Quarterly, 5, 424–434.
21. Kantor, I., & Solodovnikov, A. (1989). Hypercomplex numbers, Springer-Verlag, New York.
22. Kılıç, E., & Tan, E. (2015). On binomial sums for the general second order linear recurrence. Integers, 10(6), 801–806.
23. Kılıç, E., & Tan, E. (2009). More general identities involving the terms of {Wn(a, b; p, q)}. Ars Combinatoria, 93, 459–461.
24. Köme, S., Köme, C., & Yazlik, Y. (2021). Dual-complex generalized k-Horadam numbers. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1), 117–129.
25. Majernik, V. (1996). Multicomponent number systems. Acta Physica. Polonica A 90(3), 491–498.
26. Messelmi, F. (2015). Dual-complex numbers and their holomorphic functions. Preprint. HAL ID: hal-01114178.
27. Mező, I. (2009). Several generating functions for second-order recurrence sequences, Journal of Integer Sequences, 12, Article 09.3.7.
28. Ömür, N., & Koparal, S. (2020). On hyper-dual generalized Fibonacci numbers. Notes on Number Theory and Discrete Mathematics, 26(1), 191–198.
29. Pennestri, E., & Stefanelli, R. (2007). Linear algebra and numerical algorithms using dual numbers. Multibody System Dynamics, 18(3), 323–344.
30. Rayaguru, S. G., Savin, D., & Panda, G. K. (2019). On some Horadam symbol elements. Notes on Number Theory and Discrete Mathematics, 25(2), 91–112.
31. Study, E. (1903). Geometrie der dynamen: Die zusammensetzung von kraften und verwandte gegenstande der geometrie bearb. Leipzig, B.G. Teubner. 1903.
32. Sobczyk, G. (1995). The hyperbolic number plane. The College Mathematics Journal, 26(4), 268–280.
33. Tan, E., Ait-Amrane, N. R., & Gök, I. (2021). Hyper-Dual Horadam quaternions. Preprints, 2021030435.
34. Taşkara, N., Uslu, K., Yazlık, Y., & Yılmaz, N. (2011). The construction of Horadam numbers in terms of the determinant of tridiagonal matrices. AIP Conference Proceedings, 1389(1), 367–370.
35. Yaglom, I. M. (1968). Complex numbers in geometry. Academic Press, New York.
36. Yaglom, I. M. (1979). A simple non-Euclidean geometry and its physical basis. SpringerVerlag, New York.
37. Yazlık, Y., & Taskara, N. (2012). A note on generalized k-Horadam sequence. Computers & Mathematics with Applications, 63(1), 36–41.