ON HYPER COMPLEX NUMBERS WITH HIGHER ORDER BALANCING NUMBERS COMPONENTS

Year 2026, Volume: 16 Issue: 2, 204 - 213, 03.02.2026

Ritanjali Mohanty , Hrishikesh Mahato

https://izlik.org/JA23JS67RE

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

Hyper complex numbers , Higher-order balancing numbers , Recurrence relation

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.

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