On split Narayana and Narayana–Lucas hybrid quaternions

Pankaj Kumar and Shilpa Kapoor
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 696–717
DOI: 10.7546/nntdm.2025.31.4.696-717
Full paper (PDF, 268 Kb)

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Authors and affiliations

Pankaj Kumar
Srinivasa Ramanujan Department of Mathematics, Central University of Himachal Pradesh
Dharamshala, Shahpur Parisar, Shahpur – 176206, India

Shilpa Kapoor
Srinivasa Ramanujan Department of Mathematics, Central University of Himachal Pradesh
Dharamshala, Shahpur Parisar, Shahpur – 176206, India

Abstract

In this paper, we introduce the novel concepts of split Narayana quaternions and split Narayana–Lucas quaternions within the innovative framework of hybrid numbers. We explore their deep connections with Narayana and Narayana–Lucas quaternions, uncovering new perspectives in this mathematical domain. Furthermore, we establish several fundamental properties, including recurrence relations, Binet formulas, generating functions, exponential generating functions, and other significant identities associated with these newly defined quaternions. Finally, to better illustrate these theoretical findings, we also provide a numerical simulation of split Narayana quaternions and split Narayana–Lucas hybrid quaternions.

Keywords

• Narayana quaternions
• Narayana–Lucas
• Hybrid number
• Split quaternions

2020 Mathematics Subject Classification

• 11B37
• 11B39
• 11B50
• 20G20

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Manuscript history

• Received: 12 March 2025
• Revised: 9 September 2025
• Accepted: 8 October 2025
• Online First: 10 October 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).