For $n=0,1,2, \ldots$, the degree $n$ polynomial $C_{n}^{\alpha}(x)$ satisfies the differential equation

$$\left(1-x^{2}\right) y^{\prime \prime}-(2 \alpha+1) x y^{\prime}+n(n+2 \alpha) y=0$$

where $\alpha$ is a real, positive parameter. Show that, when $m \neq n$,

$$\int_{a}^{b} C_{m}^{\alpha}(x) C_{n}^{\alpha}(x) w(x) d x=0$$

for a weight function $w(x)$ and values $a<b$ that you should determine.

Suppose that the roots of $C_{n}^{\alpha}(x)$ that lie inside the domain $(a, b)$ are $\left\{x_{1}, x_{2}, \ldots, x_{k}\right\}$, with $k \leqslant n$. By considering the integral

$$\int_{a}^{b} C_{n}^{\alpha}(x) \prod_{i=1}^{k}\left(x-x_{i}\right) w(x) d x$$

show that in fact all $n$ roots of $C_{n}^{\alpha}(x)$ lie in $(a, b)$.