We consider the set $E_2$ of functions from $]-1,1[$ to $\mathbb{R}$ of the form $$t \mapsto \sum_{n=0}^{+\infty} a_n t^n$$ where $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ and $\sum (a_n)^2$ is convergent. For $f, g \in E_2$, we set $$\langle f, g \rangle = \sum_{n=0}^{+\infty} a_n b_n \quad \text{where } f : t \mapsto \sum_{n=0}^{+\infty} a_n t^n \text{ and } g : t \mapsto \sum_{n=0}^{+\infty} b_n t^n.$$ Deduce that $E_2$ is a reproducing kernel Hilbert space and specify its kernel.

We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. It has been shown that $S(x) = W(x)$ for all $x \in ]-R,R[$. Does this result remain true on $[-R, R]$?

We assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We further set $\mathbf{f} = (1 + w)\delta - w\mathbf{1}$.
Using the notations of Dirichlet series given in subsection I.E, express, for values of the real $s$ to be specified, $L_{\mathbf{f}}(s)$ in terms of $w$ and $L_{\mathbf{1}}(s)$.

Using the notations of Dirichlet series given in subsection I.E, and with $w = \frac{1}{\lambda-1}$ and $\mathbf{f} = (1+w)\delta - w\mathbf{1}$, express, for values of the real $s$ to be specified, $L_{\mathbf{f}}(s)$ in terms of $w$ and $L_{\mathbf{1}}(s)$.

We assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We further set $\mathbf{f} = (1 + w)\delta - w\mathbf{1}$. We denote $\log_2$ the logarithm function in base 2, defined by $\log_2(x) = \frac{\ln(x)}{\ln(2)}$ for all real $x > 0$.
Show that, for $s$ real sufficiently large,
$$\frac{1}{L_{\mathbf{f}}(s)} = 1 + \sum_{m=2}^{\infty} m^{-s} \sum_{k=1}^{\lfloor \log_2 m \rfloor} w^k D_k(m)$$
where $D_k(m)$ is the number of ways to decompose the integer $m$ into a product of $k$ factors greater than or equal to 2, the order of these factors being important.

We denote $\log_2$ the logarithm function in base 2, defined by $\log_2(x) = \frac{\ln(x)}{\ln(2)}$ for all real $x > 0$. With $w = \frac{1}{\lambda-1}$ and $\mathbf{f} = (1+w)\delta - w\mathbf{1}$, show that, for $s$ real sufficiently large,
$$\frac{1}{L_{\mathbf{f}}(s)} = 1 + \sum_{m=2}^{\infty} m^{-s} \sum_{k=1}^{\left\lfloor \log_2 m \right\rfloor} w^k D_k(m)$$
where $D_k(m)$ is the number of ways to decompose the integer $m$ into a product of $k$ factors greater than or equal to 2, the order of these factors being important.

We assume that $\lambda$ is a real distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We denote $\log_2$ the logarithm function in base 2. For $n \geq 1$, we set $S_k(n) = \sum_{m=2}^{n} D_k(m)$. Deduce from the previous question that
$$\chi_n(\lambda) = (\lambda - 1)^n - \sum_{k=1}^{\lfloor \log_2 n \rfloor} (\lambda - 1)^{n-k-1} S_k(n).$$

We denote $\log_2$ the logarithm function in base 2. For $n \geq 1$, we set $S_k(n) = \sum_{m=2}^{n} D_k(m)$ where $D_k(m)$ is the number of ways to decompose the integer $m$ into a product of $k$ factors greater than or equal to 2, the order of these factors being important. Deduce from the previous question that
$$\chi_n(\lambda) = (\lambda - 1)^n - \sum_{k=1}^{\left\lfloor \log_2 n \right\rfloor} (\lambda - 1)^{n-k-1} S_k(n).$$

Let $p \in \mathbb{N}$. Show that the sequence with general term $u_n = \binom{n}{p}$ is hypergeometric.

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 $s$.
Deduce that $$\zeta(s)^{-1} = \lim_{n \rightarrow +\infty} \prod_{k=1}^{n}\left(1 - p_k^{-s}\right).$$

Prove that the set of sequences satisfying relation $$P(n) u_n = Q(n) u_{n+1}$$ with $$P(X) = X(X-1)(X-2) \quad \text{and} \quad Q(X) = X(X-2)$$ is a vector space for which we will specify a basis and the dimension.

Give without proof the value of $C _ { 3 }$ and represent all Dyck paths of length 6.

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with: $$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$ Show that: $$I = 4\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}$$ You may use the result from question 2.13.

We set $C = \exp\left(\frac{I}{2\pi}\right)$ with $I = 4\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}$. The calculator gives: $$\begin{aligned} \exp\left(\frac{2}{\pi}\sum_{k=0}^5 \frac{(-1)^k}{(2k+1)^2}\right) &\approx 1{,}78774486868 \\ \exp\left(\frac{2}{\pi}\sum_{k=0}^6 \frac{(-1)^k}{(2k+1)^2}\right) &\approx 1{,}79449196958 \end{aligned}$$ Can we deduce the rounding of $C$ to $10^{-2}$ precision? If yes, give the value of this rounding. In any case, justify the answer properly.

For $n \in \mathbb{N}$, calculate $\left\|T_n\right\|_{L^\infty([-1,1])}$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

For $n \in \mathbb{N}$, calculate $\left\|T_n\right\|_{L^\infty([-1,1])}$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

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 $s$.
We denote $r(n)$ the number of divisors $d \geqslant 1$ of $n$. Show that the series $\sum_{n=1}^{+\infty} r(n) n^{-s}$ converges and that its sum equals $\zeta(s)^2$.

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 $s$. If $n \in \mathbb{N}^*$, we set $g(n) = r_1(n) - r_3(n)$ where $r_i(n) = \operatorname{Card}\{d \in \mathbb{N} : d \equiv i [4] \text{ and } d \mid n\}$.
Deduce that the series $\sum_{n=1}^{+\infty} g(n) n^{-s}$ converges.

Using question 4, show $$\forall n \in \mathbb { N } , \quad C _ { n + 1 } = \sum _ { r = 0 } ^ { n } C _ { r } C _ { n - r } .$$

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 the sequence of functions $\left(x \mapsto \prod_{k=1}^{n} p_k^{\nu_{p_k}(x)}\right)_{n \geqslant 1}$ from $\mathbb{N}^*$ to $\mathbb{N}^*$ converges pointwise to the identity function.

Using the random variable $T$, show that the series $\sum _ { n \geqslant 0 } \frac { C _ { n } } { 4 ^ { n } }$ converges.

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.

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

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 $s$. We recall that $\chi_4(2n) = 0$ and $\chi_4(2n-1) = (-1)^{n-1}$ for $n \in \mathbb{N}^*$. We set $g(n) = r_1(n) - r_3(n)$ where $r_i(n) = \operatorname{Card}\{d \in \mathbb{N} : d \equiv i [4] \text{ and } d \mid n\}$.
Deduce that the series $$\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^s}$$ is convergent and that its sum equals $E(g(X))$.

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 $n$, $[a]_{n+1} = a[a+1]_n$.