The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
By noting that for every real $\theta$, we have $e^{in\theta} = \left(e^{i\theta}\right)^n$, show that $$\forall n \in \mathbb{N}, \quad T_n = \sum_{0 \leqslant k \leqslant n/2} \binom{n}{2k} \left(X^2 - 1\right)^k X^{n-2k}$$
Write in Maple or Mathematica language a function $T$ taking as argument a natural integer $n$ and returning the expanded expression of the polynomial $T_n$.

The Chebyshev polynomials of the first kind $(T_n)_{n \in \mathbb{N}}$ are defined by $T_n(\cos\theta) = \cos(n\theta)$.
Show that the sequence $(T_n)_{n \in \mathbb{N}}$ satisfies the recurrence relation $$\forall n \in \mathbb{N}, \quad T_{n+2} = 2X T_{n+1} - T_n$$
Deduce, for every natural integer $n$, the degree and the leading coefficient of $T_n$. Find this result again with the expression from question I.A.2.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with the polynomial $P$ under composition.
Let $\alpha \in \mathbb{C}$ and let $Q$ be a non-constant complex polynomial that commutes with $P_\alpha$. Show that $Q$ is monic.

We denote by $n$ the integer part of $\frac{N}{2}$. We continue the study of the polynomial $R_{N}$ (the even polynomial in $B_N$ minimising $L$).
Show that $\deg R_{N} = 2n$.

Let $n \geqslant 1$ and $m \geqslant 1$ be two integers and $\alpha _ { 1 } , \ldots , \alpha _ { n } , \beta _ { 1 } , \ldots , \beta _ { m }$ be complex numbers. We define: $$\begin{aligned} & A ( X ) = \left( X - \alpha _ { 1 } \right) \left( X - \alpha _ { 2 } \right) \cdots \left( X - \alpha _ { n } \right) \\ & B ( X ) = \left( X - \beta _ { 1 } \right) \left( X - \beta _ { 2 } \right) \cdots \left( X - \beta _ { m } \right) \end{aligned}$$
Show that if $A ( X )$ and $B ( X )$ have rational coefficients, then the polynomials $$\prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } \beta _ { j } \right) \text { and } \prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } - \beta _ { j } \right)$$ also have rational coefficients.

Let $n \geqslant 1$ and $m \geqslant 1$ be two integers and $\alpha _ { 1 } , \ldots , \alpha _ { n } , \beta _ { 1 } , \ldots , \beta _ { m }$ be complex numbers. We define: $$\begin{aligned} & A ( X ) = \left( X - \alpha _ { 1 } \right) \left( X - \alpha _ { 2 } \right) \cdots \left( X - \alpha _ { n } \right) \\ & B ( X ) = \left( X - \beta _ { 1 } \right) \left( X - \beta _ { 2 } \right) \cdots \left( X - \beta _ { m } \right) . \end{aligned}$$ Show that if $A ( X )$ and $B ( X )$ have rational coefficients, then the polynomials $$\prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } \beta _ { j } \right) \quad \text { and } \quad \prod _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { m } \left( X - \alpha _ { i } - \beta _ { j } \right)$$ also have rational coefficients.

Let $P$ be an irreducible Hurwitz polynomial in $\mathbf{R}[X]$ with positive leading coefficient. Prove that all coefficients of $P$ are strictly positive.

Until the end of part A, we assume that all roots of $p$ are stable and have multiplicity 1.
Let $h$ be the polynomial of degree $n$ defined by $h(X) = X p'$, where $p'$ is the derivative polynomial of $p$. We denote by $h_0$ and $(p')_0$ the reciprocal polynomials of $h$ and $p'$ respectively.
Show that $h = np - \lambda (p')_0$, then that $h_0 = \lambda(np - Xp')$.