About Legendre’s polynomials

Let us define, on the set $\mathbb{R}[X]$ of real polynomials, the endomorphism

$$u\colon P\longmapsto \frac{\mathrm{d}}{\mathrm{d}X}\left( (1-X^2) P'(X)\right)$$

as well as the polynomials

$$\forall n\in\mathbb{N}, L_n(X):=c_n\frac{\mathrm{d}}{\mathrm{d}X}\left((X^2-1)^n\right)$$

called Legendre’s polynomials, where $c_n$ is a normalization constant such that $L_n$ is monic. Finally, one define the scalar product
$$\forall P,Q\in\mathbb{R}[X], \langle P,Q\rangle :=\int_{-1}^1 P(x)\,Q(x)\,\mathrm{d}x.$$

1. Determine the degree of $L_n$ for all $n\in\mathbb{N}$ and the value of $c_n$.
2. Show that u is symmetric, meaning that $\langle u(P), Q\rangle =\langle P,u(Q)\rangle$ for all $P,Q\in\mathbb{R}[X]$.
3. a) Show that the Legendre polynomials are all eigenvectors or $P$ with specific eigenvalues that you will compute.
4. b) Deduce that $(L_n)_n$ is a basis of $\mathbb{R}[X]$. What can be said about $\langle L_n, Q\rangle$ if $\mathrm{deg}(Q)<n$ ?
5. c) Show, using 3) a), that $L_n(-X)$ is proportional to $L_n(X)$, then determine the constant.
6. Show that $L_n$ satisfies the relation

$$XL_n'(X)=\frac{c_n}{c_{n-1}} L_{n-1}'(X)+nL_n(X)$$

for all $n\in\mathbb{N}^*$.
5) Use all the previous questions in order to determine $L_n(1), L_n(-1)$ and finally the norm $\Vert L_n\Vert_2^2:=\langle L_n, L_n\rangle$.