The Hadamard-type k-step Pell sequences

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
Full paper (PDF, 239 Kb)

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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

References

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Manuscript history

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