Insights Into the Lucas \(Q\)-Matrix and Its Properties

Abstract

This work proves that the equality
\begin{align*}
\begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix}^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}
\end{align*}
holds for all integer \( n \), where \( \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix} \) is the Lucas \( Q \)-matrix introduced by Köken and Bozkurt [2]. While these authors established the case for natural \(n\) using induction, extending the result to integer exponents is considerably more challenging. One cannot simply assume that the formula holds for an arbitrary integer \(n\), since, although Fibonacci and Lucas numbers are defined for negative indices, the proof by mathematical induction does not automatically extend to this setting. To achieve this, we developed several key prerequisites, including new relationships between the Fibonacci and Lucas \( Q \)-matrices, such as \(\begin{bmatrix}L_{n+1} & L_{n} \\ L_{n} & L_{n-1}\end{bmatrix}=\begin{bmatrix}3 & 1 \\ 1 & 2\end{bmatrix}\cdot\begin{bmatrix}F_{n} & F_{n-1} \\ F_{n-1} & F_{n-2}\end{bmatrix}=Q_L \cdot Q_F^{n-1}\), \(n \in \mathbb{Z}\). Additionally, we demonstrate several previously known properties, but in a different way, using our properties. Because this text is also a survey article, we adopted a didactic, step-by-step approach, minimizing omissions whenever possible.