Let $\ell \geq 0$ be an integer. Show that there exists a polynomial $P_\ell \in \mathbf{Q}[x]$ of degree $< r(\ell+1)$ satisfying
$$\sum_{n=0}^{\infty} n(n-1)\cdots(n-\ell+1)\, u_{n-\ell}\, x^n = \frac{P_\ell(x)}{\left(1 - s_1 x - \cdots - s_r x^r\right)^{\ell+1}}.$$