Fügen Torunbalcı Aydın

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 3, Pages 184–193

DOI: 10.7546/nntdm.2021.27.3.184-193

**Full paper (PDF, 177 Kb)**

## Details

### Authors and affiliations

Fügen Torunbalcı Aydın

*Yildiz Technical University
Faculty of Chemical and Metallurgical Engineering
Department of Mathematical Engineering
Davutpasa Campus, 34220, Esenler, Istanbul, Turkey*

### Abstract

The aim of this work is to consider the Pauli–Fibonacci quaternions and to present some properties involving this sequence, including the Binet’s formula and generating functions. Furthermore, the Honsberger identity, the generating function, d’Ocagne’s identity, Cassini’s identity, Catalan’s identity for these quaternions are given. The matrix representations for Pauli–Fibonacci quaternions are introduced.

### Keywords

- Pauli matrix
- Pauli quaternion
- Fibonacci number
- Fibonacci quaternion
- Pauli–Fibonacci quaternion

### 2020 Mathematics Subject Classification

- 11R52
- 20G20

### References

- Adler, S. L. (1995). Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York.
- Akyigit, M., Kosal, H., & Tosun, M. (2013). Split Fibonacci quaternions. Advances in Applied Clifford Algebras, 23(3), 535–545.
- Altmann, S. L. (2005). Rotations, Quaternions, and Double Groups, Dover Books on Mathematics, Reprint Edition.
- Arfken, G. B., & Weber, H. J. (2001). Mathematical Methods for Physicists (5th ed.). Harcourt/Academic Press, San Francisco.
- Cahay, M., Purdy, G. B., & Morris, D. (2019). On the quaternion representation of the Pauli spinor of an electron. Physica Scripta, 94(8), Article No. 085205.
- Clifford, W. K. (1873). Preliminary sketch of bi-quaternions. Proceedings of the London Mathematical Society, 64(4), 381–395.
- Condon, E. U. & Morse, P. M. (1931) Quantum mechanics of collision processes I. Scattering of particles in a definite force field. Reviews of Modern Physics, 3(1), 43.
- González-Díaz, F. R., & García-Salcedo, R. (2017). The phenomenon of half-integer spin, quaternions, and Pauli matrices. Revista de Matemática Teoría y Aplicaciones, 24(1), 45–60.
- Halıcı, S. (2012). On Fibonacci Quaternions. Advances in Applied Clifford Algebras, 22(2), 321–327.
- Halıcı, S. (2013). On Complex Fibonacci Quaternions. Advances in Applied Clifford Algebras, 23(1), 105–112.
- Hamilton, W. R. (1866). Elements of Quaternions. Longmans, Green and Co., London.
- Horadam, A. F. (1963). Complex Fibonacci Numbers and Fibonacci Quaternions. The American Mathematical Monthly, 70(3), 289–291.
- Horadam, A. F. (1993). Quaternion Recurrence Relations, Ulam Quarterly, 2(2), 23–33.
- 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, 201–210.
- Kim, J. E. (2017). A Representation of de Moivre’s formula over Pauli-quaternions. The Fibonacci Quarterly, 9(2), 145–151.
- Koshy, T. (2001). Fibonacci and Lucas Numbers with Applications. John Wiley and Sons, New York – Toronto.
- Longe, P. (1966). The properties of the Pauli matrices A, B, C and the conjugation of charge, Physica, 32(3), 603–610.
- Nurkan, K. S., & Güven, A. I. (2015) Dual Fibonacci Quaternions. Advances in Applied Clifford Algebras, 25(2), 403–414.
- Ozdemir, M., & Ergin, A. A. (2006). Rotations with unit timelike quaternions in Minkowski 3-space. Journal of Geometry and Physics, 56(2), 322–336.
- Silberstein, L. (1912). LXXVI. Quaternionic form of relativity. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23(137), 790–809.
- Swammy, M. N. S. (1973). On generalized Fibonacci quaternions. The Fibonacci Quarterly, 11(5), 547–550.
- Vajda, S. (1989). Fibonacci and Lucas Numbers, and the Golden Section. Ellis Horwood Limited Publ., England.
- Hoggatt, V. E. (1969). Fibonacci and Lucas Numbers. Houghton Mifflin, Boston.
- Yuce, S., & Aydın, F. T. (2016). A new aspect of dual Fibonacci quaternions. Advances in Applied Clifford Algebras, 26(2), 873–884.

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## Cite this paper

Aydın, F. T. (2021). Pauli–Fibonacci quaternions. *Notes on Number Theory and Discrete Mathematics*, 27(3), 184-193, DOI: 10.7546/nntdm.2021.27.3.184-193.