We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ We define for any natural integer $n$ the polynomial $R_n$ as follows $$R_n(X) = (X^2 - 1)^n$$ and we now set $$L_n(X) = \frac{1}{2^n n!} R_n^{(n)}(X)$$ Let $n \in \mathbb{N}$. (a) What is the degree of the polynomial $L_n$? Express $M(L_n)$ in terms of $L_n$. (b) Show that if $n \geq 1$ then $$\forall P \in \mathbb{R}_{n-1}[X], \quad (L_n \mid P) = 0$$ (c) Show that for every integer $k$ such that $0 \leq k \leq n$, we have $$L_n^{(k)}(1) = \frac{(n+k)!}{(n-k)!} \frac{1}{k! \, 2^k}$$ (d) Show that for every natural integer $k$, we have $$S\left(L_n, L_n^{(2k+1)}\right) = 2 L_n^{(2k+1)}(1)$$
We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product
$$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$
We define for any natural integer $n$ the polynomial $R_n$ as follows
$$R_n(X) = (X^2 - 1)^n$$
and we now set
$$L_n(X) = \frac{1}{2^n n!} R_n^{(n)}(X)$$
Let $n \in \mathbb{N}$.\\
(a) What is the degree of the polynomial $L_n$? Express $M(L_n)$ in terms of $L_n$.\\
(b) Show that if $n \geq 1$ then
$$\forall P \in \mathbb{R}_{n-1}[X], \quad (L_n \mid P) = 0$$
(c) Show that for every integer $k$ such that $0 \leq k \leq n$, we have
$$L_n^{(k)}(1) = \frac{(n+k)!}{(n-k)!} \frac{1}{k! \, 2^k}$$
(d) Show that for every natural integer $k$, we have
$$S\left(L_n, L_n^{(2k+1)}\right) = 2 L_n^{(2k+1)}(1)$$