New summation identities of hyperbolic k-Fibonacci and k-Lucas quaternions

A. D. Godase
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 3, Pages 570–582
DOI: 10.7546/nntdm.2025.31.3.570-582
Full paper (PDF, 199 Kb)

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Authors and affiliations

A. D. Godase
Department of Mathematics, V. P. College Vaijapur
Aurangabad, 423701, Maharashtra, India

Abstract

In this paper, we introduce a set of identities involving hyperbolic -Fibonacci quaternions and -Lucas quaternions. Moreover, we derive summation identities for hyperbolic -Fibonacci and -Lucas quaternions by utilizing established properties of -Fibonacci and -Lucas numbers. These findings add valuable insight into the relationships between these quaternion sequences and offer valuable insights into their properties.

Keywords

• Fibonacci quaternion
• Lucas quaternion
• -Fibonacci quaternion
• -Lucas quaternion

2020 Mathematics Subject Classification

• 11B39
• 11B37
• 11B52

References

1. Bolat, C., & Köse, H. (2010). On the properties of -Fibonacci numbers. International Journal of Contemporary Mathematical Sciences, 5(22), 1097–1105.
2. Falcón, S., & Plaza, Á (2007). The -Fibonacci sequence and the Pascal -triangle. Chaos, Solitons & Fractals, 33(1), 38–49.
3. Godase, A. D. (2019). Properties of -Fibonacci and k-Lucas octonions. Indian Journal of Pure and Applied Mathematics, 50(4), 979–998.
4. Godase, A. D. (2020). Hyperbolic -Fibonacci and -Lucas octonions. Notes on Number Theory and Discrete Mathematics, 26(3), 176–188.
5. Godase, A. D. (2021). Study of Generalized Fibonacci Sequences. Doctoral dissertation. Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University.
6. Godase, A. D. (2021). Hyperbolic -Fibonacci and -Lucas quaternions. Mathematics Student, 90(1–2), 103–116.
7. Hamilton, W. (1844). On a new species of imaginary quantities connected with a theory of quaternions. Proceedings of The Royal Irish Academy, 2, 424–434.
8. Hamilton, W. (1866). Elements of Quaternions. Longmans, Green & Co., London.
9. Horadam, A. (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70(3), 289–291.
10. Macfarlane, A. (1900). Hyperbolic quaternions. Proceedings of The Royal Society of Edinburgh, 23, 169–180.
11. Polatlı, E., & Kesim, S. (2015). A note on Catalan’s identity for the -Fibonacci quaternions. Journal of Integer Sequences, 18(8), Article 15.8.2.
12. Polatlı, E., Kızılates¸, C., & Kesim, S. (2016). On split -Fibonacci and -Lucas quaternions. Advances in Applied Clifford Algebras, 26, 353–362.
13. Ramírez, J. (2015). Some combinatorial properties of the -Fibonacci and the -Lucas quaternions. Analele ştiinţifice ale Universităţii “Ovidius” Constanţa. Seria Matematică, 23(2), 201–212.

Manuscript history

• Received: 27 February 2025
• Revised: 6 August 2025
• Accepted: 17 August 2025
• Online First: 22 August 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).