Binomial transform of the bivariate Fibonacci quaternion polynomials and its properties

Faruk Kaplan and Arzu Özkoç Öztürk
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 547–562
DOI: 10.7546/nntdm.2025.31.3.547-562
Full paper (PDF, 228 Kb)

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Authors and affiliations

Faruk Kaplan
Department of Mathematics, Faculty of Science and Arts, University of Düzce
Konuralp, Düzce, Türkiye

Arzu Özkoç Öztürk
Department of Mathematics, Faculty of Science and Arts, University of Düzce
Konuralp, Düzce, Türkiye

Abstract

The primary aim of this work is to deal with binomial transforms of bivariate Fibonacci quaternion polynomial sequence. The binomial sequence of the bivariate Fibonacci quaternion polynomial is found, and then results are obtained for the recurrence relation, generating function and Binet formula. Furthermore, different types of sums are being sought for the polynomials. While working with bivariate Fibonacci quaternion polynomial sequence, we found the general formula for all sequences with quadratic recurrence relation, which includes the formulas in general terms for binomial transform. In the last part, matrix representations are derived for the corresponding binomial sequence.

Keywords

• Bivariate polynomials
• Binomial transform
• Binet formula
• Quaternion

2020 Mathematics Subject Classification

• 11B37
• 11B83
• 11B65
• 11R52

References

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Manuscript history

• Received: 30 October 2024
• Revised: 3 August 2025
• Accepted: 11 August 2025
• Online First: 20 August 2025

Copyright information

Ⓒ 2025 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).