**A. D. Godase**

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 26, 2020, Number 3, Pages 176–188

DOI: 10.7546/nntdm.2020.26.3.176-188

**Full paper (PDF, 182 Kb)**

## Details

### Authors and affiliations

A. D. Godase

*Department of Mathematics, V. P. College Vaijapur
Aurangabad (MH), India*

### Abstract

In this paper, we introduce the hyperbolic *k*-Fibonacci and *k*-Lucas octonions. We present Binet’s formulas, Catalan’s identity, Cassini’s identity, d’Ocagne’s identity and generating functions for the k-Fibonacci and k-Lucas hyperbolic octonions.

### Keywords

- Fibonacci sequence
*k*-Fibonacci sequence*k*-Lucas sequence

### 2010 Mathematics Subject Classification

- 11B39
- 11B37

### References

- Altmann, S. L. (1986). Rotations, Quaternions, and Double Groups, Oxford University Press.
- Bolat C., & Hasan K. (2010). On the properties of
*k*-Fibonacci numbers, Int. J. Contemp. Math. Sciences, 5 (22), 1097–1105. - Cariow, A., Cariowa, G., & Knapinski, J (2015). Derivation of a low multiplicative complexity algorithm for multiplying hyperbolic octonions, arXiv preprint arXiv:1502.06250.
- Cariow, A., & Cariowa, G. (2015). A unified approach for developing rationalized algorithms for hypercomplex number multiplication, Electric Review, 91(2), 36–39.
- Catarino, P. (2014). A note on
*h*(*x*)-Fibonacci quaternion polynomials, Chaos Solitons Fractals, 77, 1–5. - Catarino, P., & Vasco, P. (2013). Some basic properties and a two-by-two matrix involving the
*k*-Pell numbers, Int. J. Math. Anal.(Ruse), 7 (45), 2209–2215. - Dhakne, M. B., & Godase, A. D. (2014). On the properties of
*k*-Fibonacci and*k*-Lucas numbers, International Journal of Advances in Applied Mathematics and Mechanics, 2 (1), 100–106. - Falcon, S. (2012). Generalized Fibonacci sequences generated from a
*k*-Fibonacci sequence, Journal of Mathematics Research, 4 (2), 97–100. - Falcon, S. (2011). On the
*k*-Lucas numbers, Int. J. Contemp. Math. Sciences, 6 (21), 1039–1050. - Falcon, S., & Plaza, A. (2007). The
*k*-Fibonacci sequence and the Pascal 2-triangle, Chaos, Solitons & Fractals, 33 (1), 38–49. - Falcon, S., & Plaza, A. (2009). On
*k*-Fibonacci numbers of arithmetic indexes, Applied Mathematics and Computation, 208 (1), 180–185. - Falcon, S., & Plaza, A. (2009). On
*k*-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals, 39 (3), 1005–1019. - Fritzer, H. P. (2001). Molecular symmetry with quaternions, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy, 57 (10), 1919–1930.
- Godase, A. D. (2019). Properties of
*k*-Fibonacci and*k*-Lucas octonions, Indian J. Pure Appl. Math., 54 (4), 979–998. - Godase, A. D. (2019). Hyperbolic
*k*-Fibonacci and*k*-Lucas Quaternions, Mathematics Student, submitted. - Goldstein, H. (1980). Classical Mechanics, Addison-Wesley Series in Physics.
- Halici, S. (2012). On Fibonacci quaternions, Adv. Appl. Clifford Algebras, 22, 321–327.
- Hamilton, W. R. (1866). Elements of Quaternions, Cambridge University Press.
- Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions, The American Mathematical Monthly, 70 (3), 289–291.
- Iyer, M. R. (1969). A note on Fibonacci quaternions, The Fibonacci Quarterly, 7 (3), 225–229.
- Iyer, M. R. (1969). Some results on Fibonacci quaternions, The Fibonacci Quarterly, 7 (2), 201–210.
- Polatli, E., & Kesim, S. (2015). On quaternions with generalized Fibonacci and Lucas number components, Advances in Difference Equations, 2015, (Article No. 169), 8 pages.
- Ramirez, J. L. (2015). Some combinatorial properties of the
*k*-Fibonacci and the*k*-Lucas quaternions, An. St. Univ. Ovidius Constanta, 23, 201–212. - Tinkham, M. (1964). Group Theory and Quantum Mechanics, McGraw-Hill Book Company.
- Tosun, M., Akyigit, M., & Kosal, H. (2013). Split Fibonacci quaternions, Adv. Appl. Clifford Algebras, 23, 535–545.
- Tosun, M., Akyigit, M., & Kosal, H. (2014). Fibonacci generalized quaternions, Adv. Appl. Clifford Algebras, 24, 631–641.
- Yazlik, Y., Yilmaz, N., & Taskara, N. (2012). On the sums of powers of
*k*-Fibonacci and*k*-Lucas sequences, Selcuk J. Appl. Math., Special Issue, 47–50.

## Related papers

- Özkan, E., & Uysal, M. (2022). On hyperbolic
*k*-Jacobsthal and*k*-Jacobsthal–Lucas octonions.*Notes on Number Theory and Discrete Mathematics*, 28(2), 318–330. - Godase, A. D. (2024). Some new properties of hyperbolic
*k*-Fibonacci and*k*-Lucas octonions.*Notes on Number Theory and Discrete Mathematics*, 30(1), 100-110.

## Cite this paper

Godase, A. D. (2020). Hyperbolic *k*-Fibonacci and *k*-Lucas octonions. *Notes on Number Theory and Discrete Mathematics*, 26 (3), 176-188, DOI: 10.7546/nntdm.2020.26.3.176-188.