We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$ Let $n \in \mathbb{N}^*$. Show that, for all $Q \in \mathbb{R}_{n-1}[X]$, $\langle Q, L_n \rangle = 0$. Hint: you may integrate by parts.
We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by:
$$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$
We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by:
$$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
Let $n \in \mathbb{N}^*$. Show that, for all $Q \in \mathbb{R}_{n-1}[X]$, $\langle Q, L_n \rangle = 0$.
Hint: you may integrate by parts.