For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$.
By enumerating the different well-parenthesized words of length 2, 4 and 6, show that $C_{1} = 1$, $C_{2} = 2$ and determine $C_{3}$.

Deduce that the power series $\sum _ { n \geqslant 0 } C _ { n } t ^ { n }$ converges uniformly on the interval $I = \left[ - \frac { 1 } { 4 } , \frac { 1 } { 4 } \right]$.

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 convention $C_{0} = 1$.
Show by a combinatorial argument that, for every integer $k \geqslant 1$, $$C_{k} = \sum_{i=0}^{k-1} C_{i} C_{k-i-1}$$ We can note that a well-parenthesized word is necessarily of the form $(m)m^{\prime}$ with $m$ and $m^{\prime}$ two well-parenthesized words, possibly empty.

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

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 $x \in ]-\frac{1}{4}, \frac{1}{4}[$, $F(x) = 1 + x(F(x))^{2}$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\forall t \in I , g ( t ) ^ { 2 } = 2 g ( t ) - 4 t$.

We recall that the sequence $\left(\left(\sum_{k=1}^{n} k^{-1}\right) - \ln(n)\right)_{n \geqslant 1}$ converges. We denote $\gamma$ its limit. Let $n \in \mathbb{N}^*$. For $x \in ]0, +\infty[$, we set $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the sequence of functions $\left(\Gamma_n\right)_{n \geqslant 1}$ converges pointwise on $]0, +\infty[$ to a function $\Gamma$ from $]0, +\infty[$ to $]0, +\infty[$.

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.

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 the function $f : \left|]-\frac{1}{4}, \frac{1}{4}[ \begin{array}{cc} & \rightarrow \\ x & \mathbb{R} \\ & \mapsto 2xF(x) - 1 \end{array}\right.$ does not vanish.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that there exists a function $\varepsilon : I \rightarrow \{ - 1,1 \}$ such that $$\forall t \in I , \quad g ( t ) = 1 + \varepsilon ( t ) \sqrt { 1 - 4 t } .$$

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that, for all $x \in ]0, +\infty[$, we have $\Gamma(x+1) = x\Gamma(x)$.

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.

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 $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Determine, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, an expression of $F(x)$ as a function of $x$.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the function $\Gamma$ is of class $\mathscr{C}^2$ and that, for all $x \in ]0, +\infty[$, $$(\ln(\Gamma))''(x) = \sum_{k=0}^{+\infty} \frac{1}{(x+k)^2}.$$

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that $\lim_{x \rightarrow +\infty} (\ln(\Gamma))''(x) = 0$.

Determine the power series expansion of the function $u \mapsto \sqrt{1-u}$. We will write the coefficients as a quotient of factorials and powers of 2.

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$ Let $f : ]0, +\infty[ \rightarrow ]0, +\infty[$ be a function of class $\mathscr{C}^2$ such that the function $\ln(f)$ is convex and satisfies $f(1) = 1$ and $f(x+1) = xf(x)$ for all $x > 0$. The function $S(x) = \ln\left(\frac{f(x)}{\Gamma(x)}\right)$ is 1-periodic and convex.
Deduce that $f = \Gamma$.

Show that, for every natural integer $n$, $$C_{n} = \frac{(2n)!}{(n+1)! \, n!}$$

Let $\Gamma$ be the Gamma function defined as the pointwise limit on $]0, +\infty[$ of $\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}$.
Show that for all $a \in ]0, +\infty[$ and $x \in ]0, +\infty[$: $$\int_0^{+\infty} \frac{t^{x-1}}{(1+t)^{x+a}} dt = \frac{\Gamma(x)\Gamma(a)}{\Gamma(x+a)}.$$ Hint: you may set, for $x \in ]0, +\infty[$, $f(x) = \frac{\Gamma(x+a)}{\Gamma(a)} \int_0^{+\infty} \frac{t^{x-1}}{(1+t)^{x+a}} dt$.

Let $\Gamma$ be the Gamma function. Show that for all $x \in ]0,1[$: $$\int_0^{+\infty} \frac{t^{x-1}}{1+t} dt = \frac{\pi}{\sin(\pi x)}.$$

Show that for all $x \in ]0,1[$: $$\frac{\pi}{\sin(\pi x)} = \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+x} + \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+1-x}.$$

Recall Stirling's formula. Deduce an asymptotic equivalent of $C _ { n }$ as $n$ tends to $+ \infty$.

We consider the case where $I = [-1,1]$ and $w(x) = 1$. Let $(p_n)_{n \in \mathbb{N}}$ be the sequence of orthogonal polynomials associated with the weight $w$ (monic, $\deg(p_n) = n$, orthogonal for $\langle f, g \rangle = \int_{-1}^1 f(x)g(x)\,\mathrm{d}x$).
Determine the first four orthogonal polynomials $(p_0, p_1, p_2, p_3)$ associated with the weight $w$.