# Applications of some special numbers obtained from a difference equation of degree three

## TL;DR

This paper explores applications of special numbers derived from a third-degree difference equation, focusing on their use in coding theory and extending these numbers to generalized quaternions.

## Contribution

It introduces generalized Pell-Fibonacci-Lucas numbers from a third-degree difference equation and extends their application to generalized quaternions, highlighting new mathematical connections.

## Key findings

- Derived special numbers from a third-degree difference equation.
- Applied these numbers to coding theory.
- Extended the numbers to generalized quaternions.

## Abstract

In this paper we present applications of some special numbers obtained from a difference equation of degree three, especially in the Coding Theory. As a particular case, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternions.