The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials

Merve Taştan, Engin Özkan and Anthony G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 148–158
DOI: 10.7546/nntdm.2021.27.2.148-158
Full paper (PDF, 802 Kb)

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Authors and affiliations

Merve Taştan
Graduate School of Natural and Applied Sciences, Erzincan Binali Yildirim University
Erzincan, Turkey

Engin Özkan
Department of Mathematics, Erzincan Binali Yildirim University
Erzincan, Turkey

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

Abstract

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized k-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini’s Identities for the families and their polynomials. Finally, we suggest avenues for further research.

Keywords

• Generalized Fibonacci polynomials
• k-Fibonacci numbers
• Generalized Lucas polynomials
• k-Lucas numbers

2020 Mathematics Subject Classification

• 11B39
• 11B83
• 11C20

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