2.II.6B

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 $(*)$.

*Typos? Please submit corrections to this page on GitHub.*