The quaternion-type cyclic-balancing sequence in groups

Nazmiye Yılmaz, Esra Kırmızı Çetinalp and Ömür Deveci
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 2, Pages 201–210
DOI: 10.7546/nntdm.2025.31.2.201-210
Full paper (PDF, 232 Kb)

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Authors and affiliations

Nazmiye Yılmaz
Department of Mathematics, Karamanoğlu Mehmetbey University
Karaman, Türkiye

Esra Kırmızı Çetinalp
Department of Engineering Sciences, Bandırma Onyedi Eylül University
Balıkesir, Türkiye

Ömür Deveci
Department of Mathematics, Kafkas University
Kars, Türkiye

Abstract

In this research, several types of definitions of the quaternion-type cyclic-balancing sequence are presented. The Cassini formula and generating function of these sequences are also obtained for all types. The quaternion-type cyclic-balancing sequences modulo the first step to transferring this topic to group theory, are examined. These sequences in finite groups are then defined. Eventually, the lengths of periods for these sequences of the generalized quaternion group are calculated.

Keywords

• Balancing sequence
• Groups
• Period
• Presentation

2020 Mathematics Subject Classification

• 11B50
• 11K31
• 20F05

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Manuscript history

• Received: 9 March 2025
• Revised: 25 April 2024
• Accepted: 26 April 2025
• Online First: 2 May 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).