# The Hadamard Products for bi-periodic Fibonacci and bi-periodic Lucas Generating matrices

## TL;DR

This paper introduces matrix representations for bi-periodic Fibonacci and Lucas polynomials, explores their properties, and examines the Hadamard products of their generating matrices, revealing new structural insights.

## Contribution

It defines new matrix forms for bi-periodic Fibonacci polynomials and investigates the properties of Hadamard products of their generating matrices, a novel approach in this area.

## Key findings

- Derived nth power and determinant formulas for the matrices.
- Established properties of Hadamard products of bi-periodic Fibonacci and Lucas matrices.
- Provided new insights into the structure of bi-periodic Fibonacci and Lucas sequences.

## Abstract

In this paper, firstly, we define the Qq-generating matrix for bi-periodic Fibonacci polynomial. And we give nth power, determinant and some properties of the bi-periodic Fibonacci polynomial by considering this matrix representation. Also, we introduce the Hadamard products for bi-periodic Fibonacci Qnq generating matrix and bi-periodic Lucas Qnl generating matrix of which entries is bi-periodic Fibonacci and Lucas numbers. Then, we investigate some properties of these products.