On Horadam quaternions by using matrix method

TL;DR

This paper introduces matrix representations for Horadam quaternions, deriving identities and binomial-sum formulas, highlighting non-commutative properties and extending known Fibonacci-like identities.

Contribution

It provides new matrix-based representations and identities for Horadam quaternions, including non-commutative analogues and binomial-sum formulas.

Findings

Derived several matrix representations for Horadam quaternions

Established identities related to quaternion multiplication

Presented binomial-sum identities for these quaternions

Abstract

In this paper, we give several matrix representations for the Horadam quaternions. We derive several identities related to these quaternions by using the matrix method. Since quaternion multiplication is not commutative, some of our results are non-commutative analogues of the well known identities for the Fibonacci-like integer sequences. Lastly, we derive some binomial-sum identities for the Horadam quaternions as an application of the matrix method.