Two solutions of the recurrence relation

$$x_{n+2}+b(n) x_{n+1}+c(n) x_{n}=0$$

are given as $p_{n}$ and $q_{n}$, and their Wronskian is defined to be

$$W_{n}=p_{n} q_{n+1}-p_{n+1} q_{n}$$

Show that

$$W_{n+1}=W_{1} \prod_{m=1}^{n} c(m)$$

Suppose that $b(n)=\alpha$, where $\alpha$ is a real constant lying in the range $[-2,2]$, and that $c(n)=1$. Show that two solutions are $x_{n}=e^{i n \theta}$ and $x_{n}=e^{-i n \theta}$, where $\cos \theta=-\alpha / 2$. Evaluate the Wronskian of these two solutions and verify $(*)$.