Nayil Kılıç

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 25, 2019, Number 1, Pages 137—149

DOI: 10.7546/nntdm.2019.25.1.137-149

**Download full paper: PDF, 294 Kb**

## Details

### Authors and affiliations

Nayil Kılıç

*Department of Mathematical Education
Hasan Ali Yücel Education Faculty, Istanbul University-Cerrahpaşa
34470, Fatih, Istanbul, Turkey
*

### Abstract

In this paper, we introduce the dual Horadam octonions, we give the Binet formula, generating function, exponential generating function, summation formula, Catalan’s identity, Cassini’s identity and d’Ocagne’s identity of dual Horadam octonions. Employing these results, we present the Binet formula, generating function, summation formula, Catalan, Cassini and d’Ocagne identities for dual Fibonacci, dual Lucas, dual Jacobsthal, dual Jacobsthal–Lucas, dual Pell and dual Pell–Lucas octonions. So we generalize results that were obtained earlier by scientists. Finally, we introduce the matrix generator for dual Horadam octonions and this generator gives the Cassini formula for the dual Horadam octonions.

### Keywords

- Generating functions
- Fibonacci and Lucas numbers and generalizations
- Octonions
- Horadam sequence

### 2010 Mathematics Subject Classification

- 05A15
- 11B39
- 17A20
- 11B83

### References

- Akkuş, I., & Keçilioğlu, O. (2015). Split Fibonacci and Lucas Octonions, Adv. Appl. Clifford Algebras, 25, 3, 517–525.
- Bremmer, M. (1999) Quantum octonions, Commun. Algebrs, 27, 2809–2831.
- Catarino, P., & Vasco, P. (2017). On Dual k-Pell Quaternions and Octonions, Mediterranean Journal of Mathematics,14:75.https://doi.org/10.1007/s00009-017-0848-3.
- Çimen, ÇB., & Ipek, A. (2017). On Jacobsthal and Jacobsthal Lucas Octonions, Mediterranean Journal of Mathematics, 14, 2.
- Clifford, W. K. (1871). Preliminary sketch of bi-quaternions, Proc. Lond. Math. Soc., 4, 1, 381–395.
- Emch, G. G. (1972). Algebraic methods in statistical mechanics and quantum field theory, Wiley Interscience, New York.
- Halıcı, S. (2015). On Dual Fibonacci Octonions, Adv. Appl. Clifford Algebras, 25, 905–914.
- Halıcı, S., & Karataş, A. (2017). Dual Horadam Octonions, arXiv preprint, arXiv:1702.08657.
- Horadam, A. F. (1965). Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J. 32, 437–446.
- Horadam, A. F. (1965) Basic properties of certain generalized sequence of numbers, The Fibonacci Quarterly, 3, 161–176.
- Ipek, A., & Arı, K. (2015). On h(x)-Fibonacci octonion polynomials, Alabama Journal of Mathematics, 39, 1–6.
- Kugo, T., & Townsend, P. (1983). Supersymmetry and the division algebras, Nuclear Physics B, 221, 357–380.
- Keçilioğlu, O., & Akkuş, I. (2015) The Fibonacci octonions, Adv. Appl. Clifford Algebras, 25, 1, 151–158.
- Liana, A. Ş, & Włoch, I. (2016). The Pell quaternions and the Pell octonions, Adv. Appl. Clifford Algebras, 26, 1, 435–440.
- Nurkan, Ş K., & G¨uven, I. A. (2015) Dual Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 25, 2, 403–414.
- Ünal, Z., Tokeşer, Ü., & Bilgici, G. (2017). Some properties of Dual Fibonacci and Dual Lucas Octonions, Adv. Appl. Clifford Algebras, 27, 1907–1916.

## Related papers

## Cite this paper

APAKılıç, N. (2019). On dual Horadam octonions. Notes on Number Theory and Discrete Mathematics, 25(1), 137-149, doi: 10.7546/nntdm.2019.25.1.137-149.

ChicagoKılıç, Nayil. “On Dual Horadam Octonions.” Notes on Number Theory and Discrete Mathematics 25, no. 1 (2019): 137-149, doi: 10.7546/nntdm.2019.25.1.137-149.

MLAKılıç, Nayil. “On Dual Horadam Octonions.” Notes on Number Theory and Discrete Mathematics 25.1 (2019): 137-149. Print, doi: 10.7546/nntdm.2019.25.1.137-149.