Recall the hypothesis $b_1 e^{a_1} + \cdots + b_r e^{a_r} = 0$ of Proposition 1. Define rational numbers $s_1, \ldots, s_r$ by the formula
$$(T - a_1) \cdots (T - a_r) = T^r - s_1 T^{r-1} - \cdots - s_{r-1} T - s_r.$$
Show the equality: for all $n \geq 0$,
$$u_{n+r} = s_1 u_{n+r-1} + \cdots + s_r u_n.$$