For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$.
Show that, for every natural integer $n$, $C_{n} \leqslant 2^{2n}$. What can we deduce for the radius of convergence of the power series $\sum C_{k} x^{k}$?

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$.
Show that, for every natural integer $n$, $C_{n} \leqslant 2^{2n}$. What can we deduce for the radius of convergence of the power series $\sum C_{k} x^{k}$?

The Pochhammer symbol is defined, for any real number $a$ and any natural integer $n$, by $$[a]_n = \begin{cases} 1 & \text{if } n = 0 \\ a(a+1)\cdots(a+n-1) = \prod_{k=0}^{n-1}(a+k) & \text{otherwise} \end{cases}$$ Let $n \in \mathbb{N}$. Give an expression of $[a]_n$

• using factorials when $a \in \mathbb{N}^*$;
• using two values of the function $\Gamma$, when $a \in D$.

Let $x \in \mathbb{R} \backslash \mathbb{Z}$. Let $m \in \mathbb{N}$ such that $m > |x|$. We set, for $n \in \mathbb{N}$ such that $n > m$: $$u_{m,n}(x) = (2n+1)\sin\left(\frac{\pi x}{2n+1}\right) \prod_{k=1}^{m}\left(1 - \frac{\sin^2\left(\frac{\pi x}{2n+1}\right)}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right)$$ and $$v_{m,n}(x) = \prod_{k=m+1}^{n}\left(1 - \frac{\sin^2\left(\frac{\pi x}{2n+1}\right)}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right).$$
Show that the sequences, indexed by $n$, $\left(u_{m,n}(x)\right)_{n > m}$ and $\left(v_{m,n}(x)\right)_{n > m}$ are convergent in $\mathbb{R}^*$.

Let $x \in \mathbb{R} \backslash \mathbb{Z}$. Let $m \in \mathbb{N}$ such that $m > |x|$. We set, for $n \in \mathbb{N}$ such that $n > m$: $$v_{m,n}(x) = \prod_{k=m+1}^{n}\left(1 - \frac{\sin^2\left(\frac{\pi x}{2n+1}\right)}{\sin^2\left(\frac{k\pi}{2n+1}\right)}\right).$$ We denote $v_m(x)$ the limit of $\left(v_{m,n}(x)\right)_{n > m}$.
Show that, for $n \in \mathbb{N}$ such that $n > m$, we have $$1 \geqslant v_{m,n}(x) \geqslant \prod_{k=m+1}^{n}\left(1 - \frac{\pi^2 x^2}{4k^2}\right)$$ and deduce that $\lim_{m \rightarrow +\infty} v_m(x) = 1$.

Deduce that, for all $x \in \mathbb{R}$, $$\sin(\pi x) = \pi x \lim_{n \rightarrow +\infty} \prod_{k=1}^{n}\left(1 - \frac{x^2}{k^2}\right).$$

Given three real numbers $a, b$ and $c$, the Gauss hypergeometric function associated with the triplet $(a, b, c)$ is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that, if $c \in D$, then $\frac{[a]_n [b]_n}{[c]_n}$ is well defined for any natural integer $n$.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, $F(x) = 1 + x(F(x))^{2}$.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Show that the power series $\sum \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$ is hypergeometric and specify associated polynomials.

Let $n$ be a non-zero natural number. Let $f$ be in $\mathcal{S}_n$. Using the result of question 13, deduce that $$\forall \theta \in \mathbb{R}, \quad |f'(\theta)| \leqslant n \|f\|_{L^\infty(\mathbb{R})}$$

Let $n$ be a non-zero natural number. Let $f \in \mathcal{S}_n$. Using the result of question 13, deduce that $$\forall \theta \in \mathbb{R}, \quad |f'(\theta)| \leqslant n \|f\|_{L^\infty(\mathbb{R})} \tag{I.4}$$

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Conversely, prove that the set of hypergeometric series associated with the polynomials obtained in the previous question is a vector space for which we will give a basis and specify the dimension.

Let $n$ be a non-zero natural number. Deduce from questions 3 and 14 that $$\forall P \in \mathbb{C}_n[X], \quad \forall x \in [-1,1], \quad \left|P'(x)\sqrt{1-x^2}\right| \leqslant n \|P\|_{L^\infty([-1,1])}$$

Let $n$ be a non-zero natural number. Deduce from questions 3 and 14 that $$\forall P \in \mathbb{C}_n[X], \quad \forall x \in [-1,1], \quad \left|P'(x)\sqrt{1-x^2}\right| \leqslant n \|P\|_{L^\infty([-1,1])}$$

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Determine the radius of convergence of the power series $\sum \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$.

Let $n$ be a non-zero natural number. Show that $$\forall Q \in \mathbb{C}_{n-1}[X], \quad |Q(1)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|Q(x)\sqrt{1-x^2}\right|.$$ One may consider $f : \theta \mapsto Q(\cos\theta)\sin\theta$ and verify that $f \in \mathcal{S}_n$.

Let $n$ be a non-zero natural number. Show that $$\forall Q \in \mathbb{C}_{n-1}[X], \quad |Q(1)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|Q(x)\sqrt{1-x^2}\right|.$$ One may consider $f : \theta \mapsto Q(\cos\theta)\sin\theta$ and verify that $f \in \mathcal{S}_n$.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that $F_{a,b,c}$ is of class $\mathcal{C}^1$ on $]-1,1[$. Calculate its derivative and express it using a Gauss hypergeometric function.

Let $n$ be a non-zero natural number. Let $R \in \mathbb{C}_{n-1}[X]$ and $t \in [-1,1]$. Show that $$|R(t)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|R(x)\sqrt{1-x^2}\right|.$$ One may consider the polynomial $S_t(X) = R(tX)$.

Let $n$ be a non-zero natural number. Let $R \in \mathbb{C}_{n-1}[X]$ and $t \in [-1,1]$. Show that $$|R(t)| \leqslant n \sup_{-1 \leqslant x \leqslant 1} \left|R(x)\sqrt{1-x^2}\right|.$$ One may consider the polynomial $S_t(X) = R(tX)$.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Justify that $F_{a,b,c}$ is of class $\mathcal{C}^\infty$ on $]-1,1[$ and express its $n$-th derivative using a Gauss hypergeometric function.

Let $n$ be a non-zero natural number. Deduce that, for all $P$ in $\mathbb{C}_n[X]$, $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$

Let $n$ be a non-zero natural number. Deduce that, for all $P$ in $\mathbb{C}_n[X]$, $$\left\|P'\right\|_{L^\infty([-1,1])} \leqslant n^2 \|P\|_{L^\infty([-1,1])}$$

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ Express the function $x \mapsto F_{\frac{1}{2}, 1, \frac{3}{2}}\left(-x^2\right)$ using usual functions.

We assume that, for every integer $k \in \mathbb{N}$, the function $x \mapsto x^k w(x)$ is integrable on $I$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, and $\langle p_i, p_j \rangle = 0$ for $i \neq j$).
Let $n \in \mathbb{N}^*$. We denote by $x_1, \ldots, x_k$ the distinct roots of $p_n$ that are in $\mathring{I}$ and $m_1, \ldots, m_k$ their respective multiplicities. We consider the polynomial $$q(X) = \prod_{i=1}^k (X - x_i)^{\varepsilon_i}, \quad \text{with } \varepsilon_i = \begin{cases} 1 & \text{if } m_i \text{ is odd} \\ 0 & \text{if } m_i \text{ is even.} \end{cases}$$
By studying $\langle p_n, q \rangle$, show that $p_n$ has $n$ distinct roots in $\mathring{I}$.