Investigation of Generalized Hybrid Fibonacci Numbers and Their Properties

Abstract

In \cite{Oz}, M. Özdemir defined a new non-commutative number system called hybrid numbers. In this paper, we define the hybrid Fibonacci and Lucas numbers. This number system can be accepted as a generalization of the complex (${\text{i}}^2=-1$), hyperbolic (${\text{h}}^2=1$) and dual Fibonacci number (${\epsilon }^2=0$) systems. Furthermore, a hybrid Fibonacci number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation $\text{ih}=-\text{hi}=\epsilon +\text{i}$. Then we used the Binet's formula to show some properties of the hybrid Fibonacci numbers. We get some generalized identities of the hybrid Fibonacci and hybrid Lucas numbers.

arXiv admin note: text overlap with arXiv:1707.05918