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 2r-ions. We give various algebraic properties of this higher-order balancing 2r -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 2r-ions, therefore establishing Cassini’s identity as a new type.