Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if $$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$ We assume that $a^2 + b = 0$ and $a \neq 0$. We denote $W$ and $W'$ the sequences $W = (a^n)_{n \in \mathbb{N}}$ and $W' = (na^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$. Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $a$, $W$ and $W'$.
Let $a$ and $b$ be two complex numbers such that $(a,b) \neq (0,0)$. We say that a complex sequence $U = (u_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $(E_{a,b})$ if
$$\forall n \in \mathbb{N}, \quad u_{n+2} = 2a u_{n+1} + b u_n$$
We assume that $a^2 + b = 0$ and $a \neq 0$. We denote $W$ and $W'$ the sequences $W = (a^n)_{n \in \mathbb{N}}$ and $W' = (na^n)_{n \in \mathbb{N}}$. Show that $U$ satisfies $E_{a,b}$ if and only if $U \in \operatorname{Vect}(W, W')$.\\
Determine $U$ satisfying $E_{a,b}$ and the initial conditions $u_0 = 0$ and $u_1 = 1$, as a function of $a$, $W$ and $W'$.