We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Determine the value of $\Gamma(n)$, for $n \in \mathbb{N}^*$.

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Show that if $p$ is a prime number such that $p \equiv 1 [4]$, we have $$E\left(g\left(p^{\nu_p(X)}\right)\right) = \frac{1}{1 - p^{-s}}.$$

Let $s > 1$ be a real number and let $X$ be a random variable taking values in $\mathbb{N}^*$ following the zeta distribution with parameter $s$.
Calculate $E\left(g\left(p^{\nu_p(X)}\right)\right)$ if $p$ is a prime number satisfying $p \equiv 3 [4]$.

The Pochhammer symbol is defined, for any real number $a$ and any natural integer $n$, by $$[a]_n = \begin{cases} 1 & \text{if } n = 0 \\ a(a+1)\cdots(a+n-1) = \prod_{k=0}^{n-1}(a+k) & \text{otherwise} \end{cases}$$ If $a$ is a negative or zero integer, justify that the sequence $\left([a]_n\right)_{n \in \mathbb{N}}$ is zero from a certain rank onwards.

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 $a \in \mathbb{R}$. Verify that, for any natural integer $n$, $[a]_{n+1} = a[a+1]_n$.

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 $a \in D$.

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 $n$.

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 $x > 0$.
Show that the function $$S : \begin{array}{ccc} ]0, +\infty[ & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \ln\left(\frac{f(x)}{\Gamma(x)}\right) \end{array}$$ is 1-periodic and convex.

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 $$x \mapsto \begin{cases} \frac{\ln(1+x)}{x} & \text{if } x \in ]-1,1[ \setminus \{0\} \\ 1 & \text{if } x = 0 \end{cases}$$ using a Gauss hypergeometric function.

We recall that $\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}} = \frac{\pi^{2k+1}}{2^{2k+2}(2k)!} E_{2k}$ where $E_{2k} = v^{(2k)}(0)$ and $v(x) = \frac{1}{\cos(x)}$. We also recall that $E(g(X)) = \sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^s}$ for a random variable $X$ following the zeta distribution with parameter $s > 1$, and $g(n) = r_1(n) - r_3(n)$.
Calculate $E(g(X))$ when $X$ is a random variable following the zeta distribution with parameter 3 then with parameter 5.

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ We admit that, in case of existence of all quantities present in the following expression, $$F_{a,b,c}(1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}.$$ Let $N \in \mathbb{N}, c \in D, a \in \mathbb{R}$ such that $c - a \in D$. Justify the existence of $F_{a,-N,c}(1)$ and prove that $$\sum_{k=0}^{N} (-1)^k \binom{N}{k} \frac{[a]_k}{[c]_k} = \frac{[c-a]_N}{[c]_N}.$$

Given three real numbers $a, b$ and $c$ with $c \in D$, the Gauss hypergeometric function is defined by $$F_{a,b,c}(x) = \sum_{n=0}^{+\infty} \frac{[a]_n [b]_n}{[c]_n} \frac{x^n}{n!}$$ We admit that, in case of existence of all quantities present in the following expression, $$F_{a,b,c}(1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}.$$ Let $(u, v) \in \mathbb{N}^2$ such that $N \leqslant \min(u, v)$. By taking $a = -u$ and $c = v - N + 1$, show Vandermonde's identity: $$\binom{u+v}{N} = \sum_{k=0}^{N} \binom{u}{k} \binom{v}{N-k}.$$

Let $f \in L^1(\mathbb{R})$, $\lambda \in \mathbb{R}_+^*$ and let $g$ be the function from $\mathbb{R}$ to $\mathbb{C}$ such that $g(x) = f(\lambda x)$ for all real $x$. Show that $g \in L^1(\mathbb{R})$ and, for all real $\xi$, express $\hat{g}(\xi)$ in terms of $\hat{f}$, $\xi$ and $\lambda$.

Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is defined on $\mathbb{R}$ and that $$\forall x \in \mathbb{R}, \quad (f*g)(x) = \int_{-\infty}^{+\infty} f(x-t)g(t)\,\mathrm{d}t = (g*f)(x)$$

Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Show that $f * g$ is bounded and that $\|f*g\|_\infty \leqslant \|f\|_1 \|g\|_\infty$.

Assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$. Let $k \in \mathbb{N}$. Show that, if $g$ is of class $\mathcal{C}^k$ and if the functions $g^{(j)}$ are bounded for $j \in \llbracket 0, k \rrbracket$, then $f * g$ is of class $\mathcal{C}^k$ and $(f*g)^{(k)} = f * (g^{(k)})$.

We classify cycles of length $k$ into three subsets:

• the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
• the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
• the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.

Show that, for every cycle $\vec{\imath}$ belonging to $\mathcal{C}_{k}$, $|\vec{\imath}| \leqslant \frac{k+1}{2}$.

We assume that $f \in L^1(\mathbb{R})$ and $g \in L^\infty(\mathbb{R})$ and we further assume that $g \in L^1(\mathbb{R})$ and $f * g \in L^1(\mathbb{R})$. Admitting that, for all real $\xi$, $$\int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} \mathrm{e}^{-\mathrm{i}x\xi} f(t) g(x-t)\,\mathrm{d}t\right)\mathrm{d}x \quad \text{and} \quad \int_{-\infty}^{+\infty} \left(\int_{-\infty}^{+\infty} \mathrm{e}^{-\mathrm{i}x\xi} f(t) g(x-t)\,\mathrm{d}x\right)\mathrm{d}t$$ exist and are equal, show that $\widehat{f*g} = \hat{f}\hat{g}$.

We classify cycles of length $k$ into three subsets:

• the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
• the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
• the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.

What can be said about $\mathcal{B}_{k}$ if $k$ is odd? Deduce that $\lim_{n \rightarrow +\infty} \mathbb{E}\left(\frac{1}{n} \sum_{i=1}^{n} \Lambda_{i,n}^{k}\right) = 0$ in this case.

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. The sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B = \left( \begin{array} { l l l } b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \end{array} \right)$ all of whose rows are equal. We denote by $P ^ { \infty }$ the row $\left( b _ { 1 } , \ldots , b _ { n } \right)$.
Prove that, $\forall i \in \llbracket 1 , n \rrbracket$, $b _ { i } > 0$.

Let $\varphi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \varphi(t) = \begin{cases} 0 & \text{if } t \leqslant 0 \\ \mathrm{e}^{-1/t} & \text{otherwise} \end{cases}$$ Show that $\varphi$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}$.
One may show that: $\forall k \in \mathbb{N}, \exists P_k \in \mathbb{R}[X], \forall t > 0, \varphi^{(k)}(t) = P_k(1/t)\mathrm{e}^{-1/t}$.

Assume that $k$ is even and that $\vec{\imath} \in \mathcal{B}_{k}$ is a cycle passing through $\frac{k}{2} + 1$ distinct vertices (i.e. $|\vec{\imath}| = \frac{k}{2} + 1$). We traverse the edges of $\vec{\imath}$ in order. To each edge of $\vec{\imath}$ we associate an opening parenthesis if this edge appears for the first time and a closing parenthesis if it appears for the second time.
Justify that we thus obtain a well-parenthesized word of length $k$.

We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. The sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B$ all of whose rows are equal to $P ^ { \infty } = \left( b _ { 1 } , \ldots , b _ { n } \right)$.
Prove that the sequence $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } } = \left( P ^ { ( 0 ) } M ^ { k } \right) _ { k \in \mathbb { N } }$ converges to $P ^ { \infty }$, regardless of the initial probability distribution $P ^ { ( 0 ) }$.

Let $\psi$ be the function defined on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \quad \psi(t) = \begin{cases} 0 & \text{if } t \notin \left]-1,1\right[ \\ \mathrm{e}^{1/(t^2-1)} & \text{otherwise.} \end{cases}$$ Show, by expressing it in terms of $\varphi$, that $\psi$ is of class $\mathcal{C}^\infty$.

We set $$\forall ( P , Q ) \in \mathbb { R } [ X ] \times \mathbb { R } [ X ] , \quad ( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$ Show that $( \cdot \mid \cdot )$ is an inner product on $\mathbb { R } [ X ]$.