Investigation of the $n$-th powers of the Fibonacci and Lucas $Q$-matrices via Lorentz product

• Grant number: 25/03393-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2025
• End date: June 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Lucas Antonio Caritá
• Grantee: Danilo Braga Lopes
• Host Institution: Instituto Federal de Educação, Ciência e Tecnologia de São Paulo (IFSP). Campus São José dos Campos. São José dos Campos , SP, Brazil

Abstract

The Fibonacci sequence $(F_n)$ and the Lucas sequence $(L_n)$ have a deep connection with certain matrices known as $Q$-matrices. The Fibonacci $Q$-matrix, $Q_F = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$, is known for the fact that its $n$-th power satisfies$Q_F^n = \begin{bmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{bmatrix}$. Similarly, the Lucas $Q$-matrix, $Q_L = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}$, satisfies\[Q_L^n = \begin{cases} 5^{\frac{n}{2}}\begin{bmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{bmatrix}, & \text{if } n \text{ is even}, \\5^{\frac{n-1}{2}}\begin{bmatrix} L_{n+1} & L_n \\ L_n & L_{n-1} \end{bmatrix}, & \text{if } n \text{ is odd}.\end{cases}\]These well-known $n$-th powers arise from the usual matrix product in $M_2(\mathbb{R})$, and several identities connecting Fibonacci and Lucas numbers can be derived from them. However, it is possible to define alternative products between $2 \times 2$ matrices, distinct from the canonical one, such as the Lorenz product.This project aims to investigate the properties of the $n$-th powers of the Fibonacci and Lucas $Q$-matrices when the usual matrix product is replaced by the Lorenz product. The goal is to understand how this change in the algebraic context affects the classical relations between these matrices and the Fibonacci and Lucas sequences, providing new insights into the interaction between matrix operations and numerical sequences. (AU)