ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS
Abstract
In this article, we define higher-order balancing numbers. Next, we employ higher-order balancing numbers to present a novel family of hyper complex numbers. These families are referred to as the higher-order balancing $2^r$-ions. We give various algebraic properties of this higher-order balancing $2^r$-ions, such as the recurrence relation, the generating function, Binet’s formula, Catalan's identity, Cassini's identity, d'Ocagne's identity and Vajda's identity and so on. Furthermore, we derive the matrix representation of the higher-order balancing $2^r$-ions, therefore establishing Cassini’s identity as a new type.
Keywords
Thanks
The authors would like to thank anonymous reviewers whose attentive reading and insightful remarks enabled us to improve the quality of our work in its present form.
References
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Details
Primary Language
English
Subjects
Algebra and Number Theory
Journal Section
Research Article
Publication Date
February 3, 2026
Submission Date
January 8, 2025
Acceptance Date
May 27, 2025
Published in Issue
Year 2026 Volume: 16 Number: 2
APA
Mohanty, R., & Mahato, H. (2026). ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS. TWMS Journal of Applied and Engineering Mathematics, 16(2), 204-213. https://izlik.org/JA23JS67RE
AMA
1.Mohanty R, Mahato H. ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS. JAEM. 2026;16(2):204-213. https://izlik.org/JA23JS67RE
Chicago
Mohanty, Ritanjali, and Hrishikesh Mahato. 2026. “ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS”. TWMS Journal of Applied and Engineering Mathematics 16 (2): 204-13. https://izlik.org/JA23JS67RE.
EndNote
Mohanty R, Mahato H (February 1, 2026) ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS. TWMS Journal of Applied and Engineering Mathematics 16 2 204–213.
IEEE
[1]R. Mohanty and H. Mahato, “ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS”, JAEM, vol. 16, no. 2, pp. 204–213, Feb. 2026, [Online]. Available: https://izlik.org/JA23JS67RE
ISNAD
Mohanty, Ritanjali - Mahato, Hrishikesh. “ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS”. TWMS Journal of Applied and Engineering Mathematics 16/2 (February 1, 2026): 204-213. https://izlik.org/JA23JS67RE.
JAMA
1.Mohanty R, Mahato H. ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS. JAEM. 2026;16:204–213.
MLA
Mohanty, Ritanjali, and Hrishikesh Mahato. “ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 2, Feb. 2026, pp. 204-13, https://izlik.org/JA23JS67RE.
Vancouver
1.Ritanjali Mohanty, Hrishikesh Mahato. ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS. JAEM [Internet]. 2026 Feb. 1;16(2):204-13. Available from: https://izlik.org/JA23JS67RE