# Dual Horadam Octonions

## TL;DR

This paper introduces a new generalization of dual quaternions and octonions using the Horadam sequence, deriving key identities and unifying second-order recurrence relations in a comprehensive framework.

## Contribution

It presents a novel generalization of dual quaternions and octonions based on the Horadam sequence, deriving fundamental identities and unifying recurrence relations.

## Key findings

- Derived Binet formula, generating function, Cassini identity, sum, and norm formulas.
- Unified second-order recurrence relations over dual quaternions and octonions.
- Established new identities for the generalized sequences.

## Abstract

In this study, we investigate Horadam sequence as generalization of linear recurrence equations of order two. By the aid of this sequence we obtain a new generalization for sequences of dual quaternions and dual octonions. Moreover, we derive some important identities such as Binet formula, generating function, Cassini identity, sum formula and norm formula by their Binet forms. We generalize all studied linear second order recurrence relations over dual octonions and quaternions in a single formula.