Research on splitting quaternions with generalized Tribonacci hybrid number components

Yanni Yang and Yong Deng
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 859–874
DOI: 10.7546/nntdm.2025.31.4.859-874
Full paper (PDF, 250 Kb)

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Authors and affiliations

Yanni Yang
School of Mathematics and Statistics, Kashi University
Kashi 844006, P.R. China

Yong Deng
School of Mathematics and Statistics, Kashi University
Kashi 844006, P.R. China

Abstract

This paper introduces the Generalized Tribonacci Hybrid Split Quaternion (GTHSQ), a novel split quaternion with coefficients derived from generalized Tribonacci hybrid numbers. This form unifies various existing number types, such as generalized hybrid Tribonacci numbers and Tribonacci numbers, offering a fresh perspective on quaternion theory. To systematically characterize GTHSQ, we develop a comprehensive mathematical framework. This includes defining GTHSQ through related number expressions, specifying its operational rules, and proving that it retains the third-order linear recurrence relation of generalized Tribonacci sequences. We derive its Binet-type formula and generating functions while exploring its core properties. Additionally, we extend classic combinatorial identities (Vajda, Catalan, Cassini) to GTHSQ, define the GTHSQ matrix, and analyze its product with the S-matrix—a generalized third-order linear recurrence sequence representation matrix—to obtain matrix and determinant expressions for GTHSQ. These findings verify the closed-form solution of GTHSQ in terms of combinatorial identities and matrix representation. Furthermore, we discuss potential applications of GTHSQ, including advancements in quaternion algebra, support for encryption algorithms in cryptography, and simplification of spatial transformations in physics, thereby providing new tools and insights for future research in quaternion theory and interdisciplinary studies.

Keywords

• Hybrid number
• Split quaternion
• Generalized Tribonacci sequence
• Hybrid splitting quaternion
• Matrix representation

2020 Mathematics Subject Classification

• 11B39
• 05A15

References

1. Adegoke, K. (2019). Basic properties of a generalized third order sequence of numbers. Preprint. Available online at: https://arxiv.org/abs/1906.00788.
2. Alagöz, A., Oral, K. H., & Yüce, S. (2012). Split quaternion matrices. Miskolc Mathematical Notes, 13(2), 223–232.
3. Cerda-Morales, G. (2017). On a generalization for Tribonacci quaternions. Mediterranean Journal of Mathematics, 14(6), Article ID 239.
4. Cerda-Morales, G. (2023). On bicomplex third-order Jacobsthal numbers. Complex Variables and Elliptic Equations, 68(1), 44–56.
5. Deng, Y. (2024). Algebraic properties of dual quaternions and split quaternions. Journal of Shanxi University (Natural Science Edition), 2024(2), 287–294 (in Chinese).
6. Erdogdu, M., & Özdemir, M. (2015). Split quaternion matrix representation of dual split quaternions and their matrices. Advances in Applied Clifford Algebras, 25(4), 787–798.
7. Kızılateş, C. (2020). A new generalization of Fibonacci hybrid and Lucas hybrid numbers. Chaos, Solitons & Fractals, 130, Article ID 109449.
8. Kızılateş, C. (2021). New families of Horadam numbers associated with finite operators and their applications. Mathematical Methods in the Applied Sciences, 44(18), 14371–14381.
9. Kızılateş, C., Catarino, P., & Tuğlu, N. (2019). On the bicomplex generalized Tribonacci quaternions. Mathematics, 7(1), Article ID 80.
10. Kızılateş, C., Du, W.-S., & Terzioğlu, N. (2024). On higher-order generalized Fibonacci hybrinomials: New properties, recurrence relations and matrix representations. Mathematics, 12(8), Article ID 1156.
11. Kızılateş, C., Polatlı, E., Terzioğlu, N., & Du, W.-S. (2025). On higher-order generalized Fibonacci hybrid numbers with q-integer components: New properties, recurrence relations, and matrix representations. Symmetry, 17(4), Article ID 584.
12. Kong, X. (2022). De Moivre’s theorem for representation matrices of hyperbolic generalized quaternions. Journal of Central China Normal University (Natural Science Edition), 56(4), 577–584.
13. Özdemir, M. (2018). Introduction to Hybrid numbers. Advances in Applied Clifford Algebras, 28(1), Article ID 11.
14. Özturk, I., & Özdemir, M. (2023). On geometric interpretations of split quaternions. Mathematical Methods in the Applied Sciences, 46(1), 408–422.
15. Özyurt, G., & Alagöz, Y. (2018). On hyperbolic split quaternions and hyperbolic split quaternion matrices. Advances in Applied Clifford Algebras, 28(5), Article ID 88.
16. Polatlı, E. (2023). Hybrid numbers with Fibonacci and Lucas hybrid number coefficients. Universal Journal of Mathematics and Applications, 6(3), 106–113.
17. Polatlı, E., Kızılateş, C., & Kesim, S. (2016). On split k-Fibonacci and k-Lucas quaternions. Advances in Applied Clifford Algebras, 26(1), 353–362.
18. Shannon, A. G., & Horadam, A. F. (1972). Some properties of third-order recurrence relations. The Fibonacci Quarterly, 10(2), 135–146.
19. Taşyurdu, Y. (2019). Tribonacci and Tribonacci–Lucas hybrid numbers. International Journal of Contemporary Mathematical Sciences, 14(4), 245–254.
20. Terzioğlu, N., Kızılateş, C., & Du, W.-S. (2022). New properties and identities for Fibonacci finite operator quaternions. Mathematics, 10(10), Article ID 1719.
21. Waddill, M. E., & Sacks, L. (1967). Another generalized Fibonacci sequence. The Fibonacci Quarterly, 5(3), 209–222.
22. Yağmur, T. (2020). A note on generalized hybrid Tribonacci numbers. Discussiones Mathematicae – General Algebra and Applications, 40(2), 187–199

Manuscript history

• Received: 11 November 2024
• Revised: 13 October 2025
• Accepted: 4 November 2025
• Online First: 19 November 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).