**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

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### 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).

## Related papers

- Karataş, A. (2022). On complex Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 28(3), 458–465. - 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. - Shannon, A. G. (2019). A note on generalized Leonardo numbers.
*Notes on Number Theory and Discrete Mathematics*, 25(3), 97–101. - 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. - 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.

## Cite this paper

İşbilir, Z., Akyiğit, M., & Tosun, M. (2023). Pauli–Leonardo quaternions. *Notes on Number Theory and Discrete Mathematics*, 29(1), 1-16, DOI: 10.7546/nntdm.2023.29.1.1-16.