On Hyperbolic Padovan and Hyperbolic Pell-Padovan Sequences

Abstract

In this article, we extend Padovan and Pell-Padovan numbers to Hyperbolic Padovan and Hyperbolic Pell-Padovan numbers, respectively. Moreover we obtain Binet-like formulas, generating functions and some identities related with Hyperbolic Padovan and Hyperbolic Pell-Padovan numbers.

Keywords

• Padovan numbers
• Pell-Padovan numbers
• Hyperbolic numbers
• Hyperbolic Padovan numbers
• Hyperbolic Pell-Padovan numbers.

References

1. 1. F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti and P. Zampetti, “The mathematics of Minkowski space-time: with an introduction to commutative hypercomplex numbers. Springer Science & Business Media, 2008.
2. 2. F. T. Aydın, “Hyperbolic Fibonacci sequence”, Universal Journal of Mathematics and Applications, 2019, 2(2), 59-64.
3. 3. S. Tas, “ The Hyperpolic Quadrapell Sequences”, Eastern Anatolian Journal of Science, 2021,Volume VII, Issue I, 25-29.
4. 4. S. Tas, “On Hyperbolic Jacobsthal-Lucas Sequence”, Fundamental Journal of Mathematics and Applications, 2022, 5(1), 16-20.
5. 5. L. Barreira, L. H. Popescu, and C. Valls, “Hyperbolic sequences of linear operators and evolution maps”, Milan Journal of Mathematics, 2016, 84(2), 203-216.
6. 6. H. Gargoubi and S. Kossentini, “f-algebra structure on hyperbolic numbers”, Advances in Applied Clifford Algebras, 2016, 26(4), 1211-1233.
7. 7. D. Khadjiev and Y. Göksal, “Applications of hyperbolic numbers to the invariant theory in two-dimensional pseudo-euclidean space”, Advances in Applied Clifford Algebras, 2016, 26(2), 645-668.
8. 8. A.E. Motter and M.A.F. Rosa, “Hyperbolic calculus”, Advances in Applied Clifford Algebras, 2016, 8(1), 109-128.

9. 9. A. Çağman, K. Polat and S. Taş, “A key agreement protocol based on group actions”, Numerical Methods for Partial Differential Equations, 2021, 37(2), 1112-1119.
10. 10. C. Voet, “The poetics of order: Dom hans van der laan's architectonic space”, Architectural Research Quarterly, 2012, 16(2), 137.
11. 11. K. Atanassov, D. Dimitrov and A. Shannon, “A remark on ψ-function and pell-padovan's sequence”, Notes on Number Theory and Discrete Mathematics, 2009, 15(2), 1-44.
12. 12. O. Deveci and E. Karaduman, “The pell sequences in finite groups”, Util. Math, 2015, 96, 263-276.
13. 13. Ö. Deveci and A.G. Shannon, “The quaternion-pell sequence”, Communications in Algebra, 2018, 46(12), 5403-5409.
14. 14. Ö. Deveci and A. G. Shannon, “The complex-type k-fibonacci sequences and their applications”, Communications in Algebra, 2020, pages 1-16.
15. 15. A.G. Shannon, P. G. Anderson and A.F. Horadam, “Properties of cordonnier, perrin and van der laan numbers”, International Journal of Mathematical Education in Science and Technology, 2006, 37(7), 825-831.
16. 16. A.G. Shannon, A.F. Horadam and P. G. Anderson, “The auxiliary equation associated with the plastic number”, Notes on Number Theory and Discrete Mathematics, 2006, 12(1), 1-12.
17. 17. S. Tas, O. Deveci, and E. Karaduman, “The fibonacci-padovan sequences in fnite groups”, Maejo International Journal of Science And Technology, 2014, 8(3), 279-287.
18. 18. G. Berzsenyi, “Gaussian fibonacci numbers”. 1977.
19. 19. A.F. Horadam, “Complex fibonacci numbers and fibonacci quaternions”, The American Mathematical Monthly, 1963, 70(3), 289-291.
20. 20. D. Taşcı, “Gaussian padovan and gaussian pell-padovan sequences”, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 2018, 67(2), 82-88.
21. 21. A. Güncan and Y. Erbil, “The q-fibonacci hyperbolic functions”, In AIP Conference Proceedings, American Institute of Physics, 2012, volume 1479, pages 946-949.