On a new generalization of Fibonacci hybrid numbers

TL;DR

This paper introduces bi-periodic Horadam hybrid numbers, generalizing classical hybrid numbers, and explores their properties, generating functions, and relationships with Fibonacci and Lucas hybrid numbers.

Contribution

It presents a new class of hybrid numbers called bi-periodic Horadam hybrid numbers, along with their generating functions, Binet formula, and interrelations with Fibonacci and Lucas hybrid numbers.

Findings

Derived the generating function for bi-periodic Horadam hybrid numbers.

Established the Binet formula for these hybrid numbers.

Explored relationships between Fibonacci and Lucas hybrid numbers.

Abstract

The hybrid numbers were introduced by Ozdemir [9] as a new generalization of complex, dual, and hyperbolic numbers. A hybrid number is defined by $k=a+bi+c\epsilon +dh$, where $a,b,c,d$ are real numbers and $% i,\epsilon ,h$ are operators such that $i^{2}=-1,\epsilon ^{2}=0,h^{2}=1$ and $ih=-hi=\epsilon +i$. This work is intended as an attempt to introduce the bi-periodic Horadam hybrid numbers which generalize the classical Horadam hybrid numbers. We give the generating function, the Binet formula, and some basic properties of these new hybrid numbers. Also, we investigate some relationships between generalized bi-periodic Fibonacci hybrid numbers and generalized bi-periodic\ Lucas hybrid numbers.