For a nonzero polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\tau(P))$ and $\operatorname{cd}(\tau(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ For a non-constant polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\delta(P))$ and $\operatorname{cd}(\delta(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.

Let $n \in \mathbb{N}^*$ and $$T _ { n } ( X ) = \sum _ { p = 0 } ^ { \lfloor n / 2 \rfloor } ( - 1 ) ^ { p } \binom { n } { 2 p } X ^ { n - 2 p } \left( 1 - X ^ { 2 } \right) ^ { p }.$$ Show that $T _ { n }$ is a polynomial of degree $n$. Explicitly state the leading coefficient of $T _ { n }$.

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, and set $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$. Show that $Q$ is a polynomial of degree at most $n - 1$.

Let $P \in \mathbb{K}[X]$. Determine the degree of $\Delta(P)$ as a function of that of $P$, where $\Delta(P) = P(X+1) - P(X)$.

Show that, for all $d \in \mathbb{N}^{*}$, $\Delta$ induces an endomorphism on $\mathbb{K}_{d}[X]$, where $\Delta(P) = P(X+1) - P(X)$.

We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
Let $n \in \mathbb { N } ^ { * }$. Justify that $P _ { n }$ is a polynomial function on $[ 0,1 ]$ of degree $n$ with coefficients in $\mathbb { Z }$.