Ahmet Kaya and Hayrullah Özimamoğlu

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 4, Pages 593–602

DOI: 10.7546/nntdm.2022.28.4.593-602

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## Details

### Authors and affiliations

Ahmet Kaya

*Departments of Mathematics, Faculty of Arts and Sciences,
Nevşehir Hacı Bektaş Veli University
Nevşehir, 50300, Turkey
*

Hayrullah Özimamoğlu

*Departments of Mathematics, Faculty of Arts and Sciences,
Nevşehir Hacı Bektaş Veli University
Nevşehir, 50300, Turkey
*

### Abstract

In this article, we generalize the well-known Gauss Pell numbers and refer to them as generalized Gauss *k*-Pell numbers. There are relationships discovered between the class of generalized Gauss *k*-Pell numbers and the typical Gauss Pell numbers. Also, we generalize the known Gauss Pell polynomials, and call such polynomials as the generalized Gauss *k*-Pell polynomials. We obtain relations between the class of the generalized Gauss *k*-Pell polynomials and the typical Gauss Pell polynomials. Furthermore, we provide matrices for the novel generalizations of these numbers and polynomials. After that, we obtain Cassini’s identities for these numbers and polynomials.

### Keywords

- Gauss Pell numbers
- Gauss Pell polynomials
- Gauss Fibonacci numbers
- Gauss Fibonacci polynomials
- Cassini’s identity

### 2020 Mathematics Subject Classification

- 11B37
- 11B39
- 11B83

### References

- Asçı, A., & Gurel, E. (2013). Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Numbers.
*Ars Combinatoria*, 111, 53–63. - Asçı, A., & Gurel, E. (2013). Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Polynomials.
*Notes on Number Theory and Discrete Mathematics*, 19(1), 25–36. - Berzsenyi, G. (1977). Gaussian Fibonacci numbers.
*The Fibonacci Quarterly*, 15(3) (1977), 233–236. - Halıcı, S., & Öz, S. (2018). On Gaussian Pell polynomials and their some properties.
*Palestine Journal of Mathematics*, 7(1), 251–256. - Halıcı, S., & Öz, S. (2016). On Some Gaussian Pell and Pell–Lucas numbers.
*Ordu University Journal of Science and Technology*, 6(1), 8–18. - Horadam, A. F. (1963). Complex Fibonacci numbers and Fibonacci quaternions.
*American Mathematical Monthly*, 70, 289–291. - Jordan, J. H. (1965). Gaussian Fibonacci and Lucas numbers.
*The Fibonacci Quarterly*, 3(4), 315–318. - Özkan, E., & Taştan, M. (2020). On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applications.
*Communications in Algebra*, 48(3), 952–960. - Özkan, E., & Taştan, M. (2021). On a new family of Gauss
*k*-Lucas numbers and their polynomials.*Asian-European Journal of Mathematics*, 14(06), Article ID 2150101. - Taş, S. (2019). A new family of
*k*-Gaussian Fibonacci numbers.*Journal of Balıkesır University Institute of Science and Technology*, 21(1), 184–189. - Taştan, M., & Özkan, E. (2021). On the Gauss
*k*-Fibonacci polynomials.*Electronic Journal of Mathematical Analysis and Applications*, 9(1), 124–130. - Yağmur, T. (2019). Gaussian Pell–Lucas Polynomials.
*Communications in Mathematics and Applications*, 10(4), 673–679.

### Manuscript history

- Received: 22 February 2022
- Revised: 21 September 2022
- Accepted: 30 September 2022
- Online First: 12 October 2022

## Related papers

- Asçı, A., & Gurel, E. (2013). Gaussian Jacobsthal and Gaussian Jacobsthal Lucas Polynomials.
*Notes on Number Theory and Discrete Mathematics*, 19(1), 25–36.

## Cite this paper

Kaya, A., & Özimamoğlu, H. (2022). On a new class of the generalized Gauss *k*-Pell numbers and their polynomials. *Notes on Number Theory and Discrete Mathematics*, 28(4), 593-602, DOI: 10.7546/nntdm.2022.28.4.593-602.