For $n \in \mathbb{N}^*$, $T_n$ denotes the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ where $F_n(x) = \cos(n \arccos x)$, and $x_{n,j}$ denotes the $j$-th zero of $T_n$ in increasing order.
Let $n \in \mathbb{N}^*$ and $x \in \mathbb{R} \setminus \{x_{n,j},\, 1 \leqslant j \leqslant n\}$. Show that: $$\frac{T_n'(x)}{T_n(x)} = \sum_{j=1}^{n} \frac{1}{x - x_{n,j}}$$

The Chebyshev polynomials of the second kind $(U_n)_{n \in \mathbb{N}}$ are defined by $$\forall n \in \mathbb{N}, \quad U_n = \frac{1}{n+1} T_{n+1}'$$ where $(T_n)_{n \in \mathbb{N}}$ are the Chebyshev polynomials of the first kind defined by $T_n(\cos\theta) = \cos(n\theta)$.
Show that $$\forall n \in \mathbb{N}, \quad \forall \theta \in \mathbb{R} \backslash \pi\mathbb{Z}, \quad U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$$

The Chebyshev polynomials of the first and second kind are defined by $T_n(\cos\theta) = \cos(n\theta)$ and $U_n(\cos\theta) = \frac{\sin((n+1)\theta)}{\sin\theta}$ for $\theta \notin \pi\mathbb{Z}$.
Show that $$\begin{cases} T_m \cdot T_n = \frac{1}{2}\left(T_{n+m} + T_{n-m}\right) & \text{for all integers } 0 \leqslant m \leqslant n \\ T_m \cdot U_{n-1} = \frac{1}{2}\left(U_{n+m-1} + U_{n-m-1}\right) & \text{for all integers } 0 \leqslant m < n \end{cases}$$

We introduce the Dickson polynomials of the first and second kind, $(D_n)_{n \in \mathbb{N}}$ and $(E_n)_{n \in \mathbb{N}}$, defined in the form of polynomial functions of two variables by $$D_0(x,a) = 2 \quad D_1(x,a) = x \quad E_0(x,a) = 1 \quad E_1(x,a) = x$$ then, for every integer $n \in \mathbb{N}$, $$D_{n+2}(x,a) = x D_{n+1}(x,a) - a D_n(x,a) \quad \text{and} \quad E_{n+2}(x,a) = x E_{n+1}(x,a) - a E_n(x,a)$$
Justify the following relation with Chebyshev polynomials $$\forall (x,a) \in \mathbb{C}^2, \quad D_n\left(2xa, a^2\right) = 2a^n T_n(x) \quad \text{and} \quad E_n\left(2xa, a^2\right) = a^n U_n(x)$$ as well as the following two relations, valid for every natural integer $n$ and every $(x,a) \in \mathbb{C}^* \times \mathbb{C}$ $$D_n\left(x + \frac{a}{x}, a\right) = x^n + \frac{a^n}{x^n} \quad \text{and} \quad \left(x - \frac{a}{x}\right) E_n\left(x + \frac{a}{x}, a\right) = \left(x^{n+1} - \frac{a^{n+1}}{x^{n+1}}\right)$$

We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We assume in this question that $U_{N}$ is even; we thus have $U_{N} \in \Pi_{n}$. In $\Pi_{n}$, the equation $P(1) = 1$ defines an affine subspace denoted $H_{n}$.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
(a) Show that $$\left\|U_{N}\right\|_{2} = \min\left\{\|P\|_{2} \mid P \in H_{n}\right\}$$
(b) Deduce that there exists a real number $\mu$ such that for all integers $0 \leqslant j \leqslant \frac{n}{2}$, we have $\left\langle U_{N}, P_{2j} \right\rangle = \mu$. (One may consider polynomials $P \in H_{n}$ of the form $U_{N} + t\left(P_{2j} - P_{2k}\right)$ with $t \in \mathbb{R}$.)
(c) Express $U_{N}$ in the basis of $P_{2j}$. Deduce that $$\frac{1}{\mu} = \sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}$$
(d) Establish in this case the formula $$a_{N} = \left(\sum_{0 \leqslant j \leqslant \frac{n}{2}} \frac{1}{g_{2j}}\right)^{-1}.$$

We denote by $n$ the integer part of $\frac{N}{2}$. We have $R_N(X) = U_N(X)^2$. We now assume that $U_{N}$ is odd.
For $j \in \mathbb{N}$, the polynomials $P_j$ are defined by $P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$, and $g_j = \int_{-1}^{1} P_j(x)^2\,dx$.
Express $a_{N}$ again in terms of the $g_{\ell}$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$.
Show that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1 - \omega_k)^2} + nP(\lambda)$$ One may apply equality (I.2) to the polynomial $X^{2n}$.

Let $n$ be a non-zero natural number. Let $P$ be in $\mathbb{C}_{2n}[X]$. For $k$ in $\llbracket 1, 2n \rrbracket$, we denote $\varphi_k = \frac{\pi}{2n} + \frac{k\pi}{n}$ and $\omega_k = \mathrm{e}^{\mathrm{i}\varphi_k}$. Show that $$\forall \lambda \in \mathbb{C}, \quad \lambda P'(\lambda) = \frac{1}{2n} \sum_{k=1}^{2n} P(\lambda\omega_k) \frac{2\omega_k}{(1-\omega_k)^2} + nP(\lambda) \tag{I.2}$$ One may apply equality (I.2) to the polynomial $X^{2n}$.

Let $n \in \mathbb{N}^*$ and $W$ be a monic polynomial of degree $n$. The objective of this subsection is to show that $$\sup _ { x \in [ - 1,1 ] } | W ( x ) | \geqslant \frac { 1 } { 2 ^ { n - 1 } }.$$ Show that $\sup _ { x \in [ - 1,1 ] } \left| T _ { n } ( x ) \right| = 1$. Deduce a monic polynomial of degree $n$ achieving equality in the above inequality.

For every integer $j \in \llbracket 1, n \rrbracket$, we denote by $f_j$ the polynomial $$f_j = a_n \prod_{k=j+1}^{n}\left(1 - \alpha_k X\right) \prod_{k=1}^{j-1}\left(X - \alpha_k\right)$$ with, according to standard conventions, $\prod_{k=n+1}^{n}(1-\alpha_k X) = \prod_{k=1}^{0}(X - \alpha_k) = 1$.
Show that if there exist two integers $i, k$ such that $1 \leq i < k \leq n$ and $\alpha_i \alpha_k = 1$, then $\alpha_i$ is a root of each polynomial $f_j$, where $j \in \llbracket 1, n \rrbracket$, and that the family $(f_1, \ldots, f_n)$ is linearly dependent.