Pauli–Leonardo quaternions

Zehra İşbilir, Mahmut Akyiğit and Murat Tosun
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 1, Pages 1–16
DOI: 10.7546/nntdm.2023.29.1.1-16
Full paper (PDF, 267 Kb)

Details

Authors and affiliations

Zehra İşbilir
Department of Mathematics, Faculty of Art and Science,
Düzce University, Düzce, 81620, Turkey

Mahmut Akyiğit
Department of Mathematics, Faculty of Science,
Sakarya University, Sakarya, 54187, Turkey

Murat Tosun
Department of Mathematics, Faculty of Science,
Sakarya University, Sakarya, 54187, Turkey

Abstract

In this study, we define Pauli–Leonardo quaternions by taking the coefficients of the Pauli quaternions as Leonardo numbers. We give the recurrence relation, Binet formula, generating function, exponential generating function, some special equalities, and the sum properties of these novel quaternions. In addition, we investigate the interrelations between Pauli–Leonardo quaternions and the Pauli–Fibonacci, Pauli–Lucas quaternions. Moreover, we create some algorithms that determine the terms of the Pauli–Leonardo quaternions. Finally, we generate the matrix representations of the Pauli–Leonardo quaternions and ℝ-linear transformations.

Keywords

• Leonardo numbers
• Pauli quaternions
• Pauli–Leonardo quaternions

2020 Mathematics Subject Classification

• 11K31
• 11R52

References

1. Adler, S. L. (1995). Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press.
2. Akyiğit, M., Kösal, H. H., & Tosun, M. (2014). Fibonacci generalized quaternions. Advances in Applied Clifford Algebras, 24(3), 631–641.
3. Alp, Y., & Koçer, E. G. (2021). Some properties of Leonardo numbers. Konuralp Journal of Mathematics, 9(1), 183–189.
4. Altmann, S. L. (2005). Rotations, Quaternions, and Double Groups. Dover Books on Mathematics, Reprint Edition.
5. Arfken, G. B., & Weber, H. J. (1999). Mathematical Methods for Physicists. American Association of Physics Teachers.
6. Cahay, M., Purdy, G. B., & Morris, D. (2019). On the quaternion representation of the Pauli spinor of an electron. Physica Scripta, 94(8), 085205.
7. Catarino, P. M. M. C., & Borges, A. (2020). On Leonardo numbers. Acta Mathematica Universitatis Comenianae, 89(1), 75–86.
8. Catarino, P. M. M. C., & Borges, A. (2020). A note on incomplete Leonardo numbers. Integers, 20, A43, 7 pages.
9. Clifford, W. K. (1871). Preliminary sketch of biquaternions. Proceedings of the London Mathematical Society, s1-4(1), 381–395.
10. Condon, E. U., & Morse, P. M. (1929). Quantum Mechanics. McGraw-Hill.
11.  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.
12. Daşdemir, A., & Bilgici, G. (2021). Unrestricted Fibonacci and Lucas quaternions.
    Fundamental Journal of Mathematics and Applications, 4(1), 1–9.
13. Dunlap, R. A. (1997). The Golden Ratio and Fibonacci Numbers. World Scientific.
14. Ercolano, J. (1979). Matrix generators of Pell sequences. Fibonacci Quarterly, 17(1), 71–77.
15. Flaut, C., & Savin, D. (2015). Quaternion algebras and generalized Fibonacci–Lucas quaternions. Advances in Applied Clifford Algebras, 25(4), 853–862.
16. 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.
17. Hamilton, W. R. (1844). III. On quaternions; or on a new system of imaginaries in algebra. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25(169), 489–495.
18. Hamilton, W. R. (1853). Lectures on Quaternions. Hodges and Smith.
19. Hamilton, W. R. (1866). Elements of Quaternions. London: Longmans, Green, & Company.
20. Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70(3), 289–291.
21. Horadam, A. F. (1961). A generalized Fibonacci sequence. The American Mathematical Monthly, 68(5), 455–459.
22. Kalman, D. (1982). Generalized Fibonacci numbers by matrix methods. Fibonacci Quarterly, 20(1), 73–76.
23. Karataş, A. (2022). On complex Leonardo numbers. Notes on Number Theory and Discrete Mathematics, 28(3), 458–465.
24. Khompungson, K., Rodjanadid, B., & Sompong, S. (2019). Some matrices in terms of Perrin and Padovan sequences. Thai Journal of Mathematics, 17(3), 767–774.
25. Kim, J. E. (2017). A representation of de Moivre’s formula over Pauli-quaternions. Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications, 9(2), 145–151.
26. Koshy, T. (2018). Fibonacci and Lucas Numbers with Applications. Volume 1. John Wiley & Sons.
27. Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications. Volume 2. John Wiley & Sons.
28. Kuhapatanakul, K. & Chobsorn, J. (2022). On the generalized Leonardo numbers. Integers, 22, A48, 7 pages.
29. Longe, P. (1966). The properties of the Pauli matrices A, B, C and the conjugation of charge. Physica, 32(3), 603–610.
30. Mangueira, M., Vieira, R., Alves, F., & Catarino, P. M. M. C. (2020). A generalização da forma matricial da sequência de Perrin. Revista Sergipana de Matemática e Educação Matemátic, 5(1), 384–392.
31. Mangueira, M. C. S., Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. C. (2022). Leonardo’s bivariate and complex polynomials. Notes on Number Theory and Discrete Mathematics, 28(1), 115–123.
32. Oduol, F., & Okoth, I. O. (2020). On generalized Fibonacci numbers. Communications in Advanced Mathematical Sciences, 3(4), 186–202.
33. Özen, K. E., & Tosun, M. (2021). Fibonacci elliptic biquaternions. Fundamental Journal of Mathematics and Applications, 4(1), 10–16.
34. Seenukul, P., Netmanee, S., Panyakhun, T., Auiseekaen, R., & Muangchan, S.-A. (2015). Matrices which have similar properties to Padovan Q-matrix and its generalized relations. Sakon Nakhon Rajabhat University Journal of Science and Technology, 7(2), 90–94.
35. Shannon, A. G. (2019). A note on generalized Leonardo numbers. Notes on Number Theory and Discrete Mathematics, 25(3), 97–101.
36. Shannon, A. G., & Deveci, Ö. (2022). A note on generalized and extended Leonardo sequences. Notes on Number Theory and Discrete Mathematics, 28(1), 109–114.
37. Shtayat, J., & Al-Kateeb, A. A. (2022). The Perrin R-matrix and more properties with an application. Journal of Discrete Mathematical Sciences and Cryptography, 25(1), 41–52.
38. Silberstein, L. (1912). LXXVI. Quaternionic form of relativity. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23(137), 790–809.
39. Sloane, N. The online encyclopedia of integer sequences. Available online at: http://oeis.org/.
40. Sokhuma, K. (2013). Padovan Q-matrix and the generalized relations. Applied
    Mathematical Sciences, 7(56), 2777–2780.
41. Sokhuma, K. (2013). Matrices formula for Padovan and Perrin sequences. Applied Mathematical Sciences, 7(142), 7093–7096.
42. Sompong, S., Wora-Ngon, N., Piranan, A., & Wongkaentow, N. (2017). Some matrices with Padovan Q-matrix property. AIP Conference Proceedings, AIP Publishing LLC, 1905(1), 030035, 6 pages.
43. Soykan, Y. (2021). Generalized Leonardo numbers. Journal of Progressive Research in Mathematics, 18(4), 58–84.
44. Soykan, Y. (2022). Special cases of generalized Leonardo numbers: Modified p-Leonardo, p-Leonardo-Lucas and p-Leonardo numbers. Preprints, 2022110045, DOI: 10.20944/preprints202211.0045.v1.
45. Torunbalcı Aydın, F. (2019). On the bicomplex k-Fibonacci quaternions. Communications in Advanced Mathematical Sciences, 2(3), 227–234.
46. Torunbalcı Aydın, F. (2021). Pauli–Fibonacci quaternions. Notes on Number Theory and Discrete Mathematics, 27(3), 184–193.
47. Vieira, R. P. M., Alves, F. R. V., & Catarino, P. M. M. C. (2019). Relações bidimensionais e identidades da sequência de Leonardo. Revista Sergipana de Matemática e Educação Matemátic, 4(2), 156–173.
48. Vieira, R. P. M., Mangueira, M. C. S., Alves, F. R. V. & Catarino, P. M. M. C. (2020). A forma matricial dos números de Leonardo. Ciência e Natura, 42(3), e100, 6 pages.
49. Vieira, R. P. M., Mangueira, M. C. S., Alves, F. R. V., & Catarino, P. M. M. C. (2021). Leonardo’s three-dimensional relations and some identities. Notes on Number Theory and Discrete Mathematics, 27(4), 32–42.
50. Waddill, M. E. (1991). Using matrix techniques to establish properties of a generalized Tribonacci sequence. In: Applications of Fibonacci Numbers, Volume 4, G. E. Bergum et al., eds. Kluwer Academic Publishers. Dordrecht, The Netherlands, pp. 299–308.
51. Waddill, M. E., & Sacks, L. (1967). Another generalized Fibonacci sequence. Fibonacci Quarterly, 5(3), 209–222.
52. Yılmaz, N., & Taskara, N. (2013). Matrix sequences in terms of Padovan and Perrin numbers. Journal of Applied Mathematics, 2013, 941673, 7 pages.
53. Yılmaz, N., & Taskara, N. (2014). On the negatively subscripted Padovan and Perrin matrix sequences. Communications in Mathematics and Applications, 5(2), 59–72.

Manuscript history

• Received: 28 July 2022
• Revised: 28 November 2022
• Accepted: 17 December 2022
• Online First: 25 January 2023

Copyright information

Ⓒ 2023 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).