Horadam $2^{k}$-Ions

Abstract

In this paper, we generalize Fibonacci quaternion, octonion, sedenion, trigintaduonion, etc. and define Horadam $2^{k}$-ions and investigate their properties. Each Horadam (such as Fibonacci, Lucas, Pell) quaternions, octonions and sedenions are Horadam $2^{k}$-ions. We also present connection to some earlier works.