Properties of hyperbolic generalized Pell numbers

Yüksel Soykan and Melih Göcen
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367-8275
Volume 26, 2020, Number 4, Pages 136–153
DOI: 10.7546/nntdm.2020.26.4.136-153
Full paper (PDF, 156 Kb)

Details

Authors and affiliations

Yüksel Soykan
Department of Mathematics, Zonguldak Bülent Ecevit University
67100, Zonguldak, Turkey

Melih Göcen
Department of Mathematics, Zonguldak Bülent Ecevit University
67100, Zonguldak, Turkey

Abstract

In this paper, we introduce the generalized hyperbolic Pell numbers over the bidimensional Clifford algebra of hyperbolic numbers. As special cases, we deal with hyperbolic Pell and hyperbolic Pell–Lucas numbers. We present Binet’s formulas, generating functions and the summation formulas for these numbers. Moreover, we give Catalan’s, Cassini’s, d’Ocagne’s, Gelin–Cesàro’s, Melham’s identities and present matrices related to these sequences.

Keywords

• Pell numbers
• Pell–Lucas numbers
• Hyperbolic numbers
• Hyperbolic Pell numbers
• Cassini identity.

2010 Mathematics Subject Classification

• 11B39
• 11B83

References

1. Akar, M., Yüce, S., & Şahin, Ş. (2018). On the Dual Hyperbolic Numbers and the Complex Hyperbolic Numbers, Journal of Computer Science & Computational Mathematics, 8(1), 1–6.
2. Baez, J. (2002). The octonions, Bull. Amer. Math. Soc., 39(2), 145–205.
3. Bicknell, M. (1975). A primer on the Pell sequence and related sequence, The Fibonacci Quarterly, 13(4), 345–349.
4. Biss, D.K., Dugger, D., & Isaksen, D.C. (2008). Large annihilators in Cayley–Dickson algebras, Communication in Algebra, 36(2), 632–664.
5. Cheng, H. H., & Thompson, S. (1996). Dual Polynomials and Complex Dual Numbers for Analysis of Spatial Mechanisms, Proc. of ASME 24th Biennial Mechanisms Conference, Irvine, CA, 19-22 August, 1996.
6. Cockle, J. (1849). On a New Imaginary in Algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 3(34), 37–47.
7. Dasdemir, A. (2011). On the Pell, Pell–Lucas and Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences, 5(64), 3173–3181.
8. Dikmen, C. M. (2019). Hyperbolic Jacobsthal Numbers, Asian Research Journal of Mathematics, 15(4), 1–9.
9. Ercolano, J. (1979). Matrix generator of Pell sequence, Fibonacci Quarterly, 17(1), 71–77.
10. Fjelstad, P., & G. Gal Sorin. (1998). n-dimensional Hyperbolic Complex Numbers, Advances in Applied Clifford Algebras, 8(1), 47–68.
11. Gökbas, H., & Köse, H. (2017). Some sum formulas for products of Pell and Pell–Lucas numbers, Int. J. Adv. Appl. Math. and Mech., 4(4), 1–4.
12. Hamilton, W. R. (1969). Elements of Quaternions, Chelsea Publishing Company, New York.
13. Horadam, A. F. (1971). Pell identities, Fibonacci Quarterly, 9 (3), 245–263.
14. Imaeda, K., & Imaeda, M. (2000). Sedenions: algebra and analysis, Applied Mathematics and Computation, 115, 77–88.
15. Jancewicz, B. (1996). The extended Grassmann algebra of R³, in Clifford (Geometric) Algebras with Applications and Engineering, Birkhauser, Boston, 389–421.
16. Kantor, I., & Solodovnikov, A. (1989). Hypercomplex Numbers, Springer-Verlag, New York.
17. Kiliç, E., & Taşçi, D. (2005). The Linear Algebra of The Pell Matrix, Boletı́n de la Sociedad Matemática Mexicana, 11(2), 163–174.
18. Kiliç, E., & Taşçi, D. (2006). The Generalized Binet Formula, Representation and Sums of the Generalized Order-k Pell Numbers, Taiwanese Journal of Mathematics, 10(6), 1661–1670.
19. Kiliç, E., & Stanica, P. (2011). A Matrix Approach for General Higher Order Linear Recurrences, Bulletin of the Malaysian Mathematical Sciences Society, 34(1), 51–67.
20. Khrennikov, A., & Segre, G. (2005). An Introduction to Hyperbolic Analysis. Available online: http://arxiv.org/abs/math-ph/0507053v2.
21. Koshy, T. (2014). Pell and Pell–Lucas Numbers with Applications, Springer, New York.
22. Kravchenko, V. V. (2009). Hyperbolic Numbers and Analytic Functions. In: Applied Pseudoanalytic Function Theory. Frontiers in Mathematics. Birkhäuser Basel, DOI: https://doi.org/10.1007/978-3-0346-0004-0 11.
23. Melham, R. (1999). Sums Involving Fibonacci and Pell Numbers, Portugaliae Mathematica, 56(3), 309–317.
24. Moreno, G. (1998). The zero divisors of the Cayley–Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana (3), 4(1), 13–28.
25. Motter A. E., & Rosa, A. F. (1998). Hyperbolic calculus, Adv. Appl. Clifford Algebra, 8(1), 109–128.
26. Ollerton, R. L. & Shannon, A. G. (1992). An extension of circular and hyperbolic functions, International Journal of Mathematical Education in Science and Technology, 23(4), 611–620.
27. Sloane, N.J.A., The on-line encyclopedia of integer sequences. Available online: http://oeis.org/.
28. Sobczyk, G. (1995). The Hyperbolic Number Plane, The College Mathematics Journal, 26(4), 268–280.
29. Sobczyk, G. (2013). Complex and Hyperbolic Numbers. In: New Foundations in Mathematics. Birkhäuser, Boston. DOI 10.1007/978-0-8176-8385-6 2.
30. Soykan, Y. (2019). Tribonacci and Tribonacci–Lucas Sedenions. Mathematics, 7(1), 74.
31. Soykan, Y. (2019). On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 20(2), 1–15.
32. Soykan, Y. (2019). On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports, 6(1), 1–18.
33. Soykan, Y. (2019). A Study of Generalized Fourth-Order Pell Sequences, Journal of Scientific Research and Reports, 25(1–2), 1–18.
34. Soykan, Y. (2019). Properties of Generalized Fifth-Order Pell Numbers, Asian Research
    Journal of Mathematics, 15(3), 1–18.
35. Soykan, Y. (2019). On Hyperbolic Numbers With Generalized Fibonacci Numbers Components, Researchgate Preprint, DOI: 10.13140/RG.2.2.19903.87207.
36. Torunbalcı Aydın, F. (2019). Hyperbolic Fibonacci Sequence, Universal Journal of Mathematics and Applications, 2(2), 59–64.