In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$. Suppose that for all $k \in \mathbb{N}$, $u_k = \mathbb{P}(X = k)$ where $X$ is a random variable taking values in $\mathbb{N}$. (a) Suppose that $X$ follows a binomial distribution with parameters $(N, p)$. Verify that $u$ satisfies condition $(C^\star)$ and find a simple expression for $u(A)$ for all $A \in \mathbb{M}_n(u)$. (b) Suppose that $X$ follows a geometric distribution with parameter $p \in ]0,1[$. Verify that $u$ satisfies condition $(C^\star)$ and show that $$u(A) = p\left(I_n - (1-p)A\right)^{-1} A$$ for every diagonalizable matrix $A \in \mathbb{M}_n(u)$.
In this part, we assume that $n \geqslant 4$. Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ satisfying the condition $(C^\star)$: $R_u > 1$.
Suppose that for all $k \in \mathbb{N}$, $u_k = \mathbb{P}(X = k)$ where $X$ is a random variable taking values in $\mathbb{N}$.\\
(a) Suppose that $X$ follows a binomial distribution with parameters $(N, p)$. Verify that $u$ satisfies condition $(C^\star)$ and find a simple expression for $u(A)$ for all $A \in \mathbb{M}_n(u)$.\\
(b) Suppose that $X$ follows a geometric distribution with parameter $p \in ]0,1[$. Verify that $u$ satisfies condition $(C^\star)$ and show that
$$u(A) = p\left(I_n - (1-p)A\right)^{-1} A$$
for every diagonalizable matrix $A \in \mathbb{M}_n(u)$.