2.II.5B

Find two linearly independent solutions of the difference equation

$X_{n+2}-2 \cos \theta X_{n+1}+X_{n}=0$

for all values of $\theta \in(0, \pi)$. What happens when $\theta=0$ ? Find two linearly independent solutions in this case.

Find $X_{n}(\theta)$ which satisfy the initial conditions

$X_{1}=1, \quad X_{2}=2,$

for $\theta=0$ and for $\theta \in(0, \pi)$. For every $n$, show that $X_{n}(\theta) \rightarrow X_{n}(0)$ as $\theta \rightarrow 0$.

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