A quantum calculus framework for Gaussian Fibonacci and Gaussian Lucas quaternion numbers

Bahar Kuloğlu
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 1–14
DOI: 10.7546/nntdm.2025.31.1.1-14
Full paper (PDF, 1037 Kb)

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Authors and affiliations

Bahar Kuloğlu
Department of Engineering Basic Sciences, Sivas Science and Technology University
Sivas, Türkiye

Abstract

In order to investigate the relationship between Gaussian Fibonacci numbers and quantum numbers and to develop both a deeper theoretical understanding in this study, q-Gaussian Fibonacci, q-Gaussian Lucas quaternions and polynomials are taken with quantum integers by bringing a different perspective. Based on these definitions, the Binet formula of these number sequences is found, and some algebraic properties, important theorems, propositions and identities related to the formula are given. Thus, new perspectives are obtained in the analysis and applications of complex systems.

Keywords

• Gaussian Fibonacci number
• Gaussian Lucas number
• Quaternion
• q-analog
• q-calculus

2020 Mathematics Subject Classification

• 11B83
• 16H05
• 05A30

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Manuscript history

• Received: 20 March 2024
• Revised: 2 December 2024
• Accepted: 23 February 2025
• Online First: 26 March 2025

Copyright information

Ⓒ 2025 by the Author.
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).