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 $(C^\star)$: $R_u > 1$.
Let $H \in \mathscr{M}_n(\mathbb{C})$ be the matrix given by $$H = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \cdots & \cdots & 0 & 0 \end{pmatrix}.$$
(a) Determine the polynomial $\varphi_H$ in this case.
(b) Let $A = H + \alpha I_n$ where $\alpha \in \mathbb{C}$ is such that $|\alpha| < R_u$. Show that $$u(A) = \sum_{k=0}^{n-1} \frac{U^{(k)}(\alpha)}{k!} H^k$$ and deduce that $$u(A) = \begin{pmatrix} U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \frac{U^{(2)}(\alpha)}{2!} & \cdots & \frac{U^{(n-1)}(\alpha)}{(n-1)!} \\ 0 & U(\alpha) & \frac{U^{(1)}(\alpha)}{1!} & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \frac{U^{(2)}(\alpha)}{2!} \\ \vdots & & \ddots & \ddots & \frac{U^{(1)}(\alpha)}{1!} \\ 0 & \cdots & \cdots & 0 & U(\alpha) \end{pmatrix}.$$