The Legendre polynomials, $P_{n}(x)$ for integers $n \geqslant 0$, satisfy the Sturm-Liouville equation

$$\frac{d}{d x}\left[\left(1-x^{2}\right) \frac{d}{d x} P_{n}(x)\right]+n(n+1) P_{n}(x)=0$$

and the recursion formula

$$(n+1) P_{n+1}(x)=(2 n+1) x P_{n}(x)-n P_{n-1}(x), \quad P_{0}(x)=1, \quad P_{1}(x)=x$$

(i) For all $n \geqslant 0$, show that $P_{n}(x)$ is a polynomial of degree $n$ with $P_{n}(1)=1$.

(ii) For all $m, n \geqslant 0$, show that $P_{n}(x)$ and $P_{m}(x)$ are orthogonal over the range $x \in[-1,1]$ when $m \neq n$.

(iii) For each $n \geqslant 0$ let

$$R_{n}(x)=\frac{d^{n}}{d x^{n}}\left[\left(x^{2}-1\right)^{n}\right]$$

Assume that for each $n$ there is a constant $\alpha_{n}$ such that $P_{n}(x)=\alpha_{n} R_{n}(x)$ for all $x$. Determine $\alpha_{n}$ for each $n$.