Ömür Deveci, Yeşim Aküzüm and Muhammad Eshaq Rashedi

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 2, Pages 339–349

DOI: 10.7546/nntdm.2022.28.2.339-349

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## Details

### Authors and affiliations

**Ömür Deveci**

*Department of Mathematics, Faculty of Science and Letters
Kafkas University, 36100 Kars, Turkey*

**Yeşim Aküzüm**

*Department of Mathematics, Faculty of Science and Letters
Kafkas University, 36100 Kars, Turkey*

**Muhammad Eshaq Rashedi**

*Department of Mathematics, Faculty of Science and Letters
Kafkas University, 36100 Kars, Turkey*

### Abstract

In this paper, we define the Hadamard-type *k*-step Pell sequence by using the Hadamard-type product of characteristic polynomials of the Pell sequence and the *k*-step Pell sequence. Also, we derive the generating matrices for these sequences, and then we obtain relationships between the Hadamard-type *k*-step Pell sequences and these generating matrices. Furthermore, we produce the Binet formula for the Hadamard-type *k*-step Pell numbers for the case that *k* is odd integers and *k* ≥ 3. Finally, we derive some properties of the Hadamard-type *k*-step Pell sequences such as the combinatorial representation, the generating function, and the exponential representation by using its generating matrix.

### Keywords

- Hadamard product
*k*-step Pell sequence- Matrix
- Representation

### 2020 Mathematics Subject Classification

- 11K31
- 11C20
- 15A15

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### Manuscript history

- Received: 7 February 2022
- Revised: 7 June 2022
- Accepted: 9 June 2022
- Online First: 10 June 2022

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## Cite this paper

Deveci, Ö., Aküzüm, Y., & Rashedi, M. E. (2022). The Hadamard-type *k*-step Pell sequences. *Notes on Number Theory and Discrete Mathematics*, 28(2), 339-349, DOI: 10.7546/nntdm.2022.28.2.339-349.