p-Analogue of biperiodic Pell and Pell–Lucas polynomials

Bahar Kuloğlu, Engin Özkan and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 29, 2023, Number 2, Pages 336–347
DOI: 10.7546/nntdm.2023.29.2.336-347
Full paper (PDF, 944 Kb)

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Authors and affiliations

Bahar Kuloğlu
Department of Mathematics, Graduate School of Natural and Applied Sciences,
Erzincan Binali Yıldırım University, Yalnızbağ Campus, 24100, Erzincan, Türkiye

Engin Özkan
Department of Mathematics, Faculty of Arts and Sciences,
Erzincan Binali Yıldırım University, Yalnızbağ Campus, 24100, Erzincan, Türkiye

Anthony G. Shannon
Warrane College, University of New South Wales
Kensington, NSW 2033, Australia

Abstract

In this study, a binomial sum, unlike but analogous to the usual binomial sums, is expressed with a different definition and termed the p-integer sum. Based on this definition, p-analogue Pell and Pell–Lucas polynomials are established and the generating functions of these new polynomials are obtained. Some theorems and propositions depending on the generating functions are also expressed. Then, by association with these, the polynomials of so-called ‘incomplete’ number sequences have been obtained, and elegant summation relations provided. The paper has also been placed in the appropriate historical context for ease of further development.

Keywords

• p-Analogue Pell
• Pell–Lucas polynomials
• Biperiodic polynomials
• Incomplete sequences

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 11B83

References

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Manuscript history

• Received: 7 March 2022
• Revised: 10 May 2023
• Accepted: 11 May 2023
• Online First: 11 May 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).