Let $\left( a _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } } , \left( b _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } } , a \in \mathbb { C }$ and $b \in \mathbb { C }$. Suppose that $\lim _ { n \rightarrow + \infty } a _ { n } = a$ and $\lim _ { n \rightarrow + \infty } b _ { n } = b$. Prove that $$\lim _ { n \rightarrow + \infty } \left( \frac { 1 } { n } \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k } \right) = a b$$

Let $\sum _ { n \geqslant 0 } a _ { n }$ and $\sum _ { n \geqslant 0 } b _ { n }$ be two series of complex numbers, convergent with respective sums $A$ and $B$. We denote $\left( c _ { n } \right) _ { n \in \mathbb { N } }$ the sequence with general term $c _ { n } = \sum _ { k = 0 } ^ { n } a _ { k } b _ { n - k }$ and $\left( C _ { n } \right) _ { n \in \mathbb { N } }$ the sequence of partial sums associated defined by $C _ { n } = \sum _ { k = 0 } ^ { n } c _ { k }$. Prove that $$\lim _ { n \rightarrow + \infty } \left( \frac { 1 } { n } \sum _ { k = 0 } ^ { n } C _ { k } \right) = A B \qquad \text{(Cauchy)}$$

Verify that the converse of (Cesàro) is not always true by exhibiting a sequence $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb{N} }$ that does not converge and such that $\left( \sigma _ { n } \right) _ { n \in \mathbb { N } }$ converges in $\mathbb { R }$.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { R } ^ { \mathbb { N } }$ and $\ell \in \mathbb { R }$. Prove that $$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } \left( u _ { n } \right) _ { n \in \mathbb { N } } \text { monotone } \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) .$$ Prove that the result holds for $\ell = + \infty$ or $\ell = - \infty$.

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. Prove that $$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } e _ { n } = o \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \qquad \text{(Weak Hardy)}$$ Hint: one may prove that for all $n \geqslant 1$, $$\sum _ { k = 0 } ^ { n } k e _ { k } = n u _ { n + 1 } - \sum _ { k = 1 } ^ { n } u _ { k }$$

Let $\left( u _ { n } \right) _ { n \in \mathbb { N } } \in \mathbb { C } ^ { \mathbb { N } }$ and $\ell \in \mathbb { C }$. The purpose of this question is to prove that $$\left( \lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell \text { and } e _ { n } = O \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \lim _ { n \rightarrow + \infty } u _ { n } = \ell \right) \qquad \text{(Strong Hardy)}$$
We suppose that $\lim _ { n \rightarrow + \infty } \sigma _ { n } = \ell$ and $e _ { n } = O \left( \frac { 1 } { n } \right)$.
(a) Let $0 \leqslant n < m$. Prove that $$\sum _ { k = n + 1 } ^ { m } u _ { k } - ( m - n ) u _ { n } = \sum _ { j = n } ^ { m - 1 } ( m - j ) e _ { j }$$
(b) Deduce that there exists a constant $C > 0$ such that for all $2 \leqslant n < m$, we have $$\left| \frac { ( m + 1 ) \sigma _ { m } - ( n + 1 ) \sigma _ { n } } { m - n } - u _ { n } \right| \leqslant C \ln \left( \frac { m - 1 } { n - 1 } \right)$$ and $$\left| u _ { n } - \ell \right| \leqslant C \ln \left( \frac { m - 1 } { n - 1 } \right) + \frac { m + 1 } { m - n } \left( \left| \sigma _ { m } - \ell \right| + \left| \sigma _ { n } - \ell \right| \right) .$$
(c) Deduce (Strong Hardy). Hint: one may take $m = 1 + \lfloor \alpha n \rfloor$ with a parameter $\alpha > 1$ to be chosen, where $\lfloor x \rfloor$ denotes the integer part of $x \in \mathbb { R }$.

Show that if $f \in \mathbf{Q}\llbracket x \rrbracket$ is the power series expansion of a rational function with rational coefficients, then $f$ is globally bounded.
(A power series $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ is globally bounded if there exist integers $A, B \geq 1$ such that $A f(Bx) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} (B^n A c_n) x^n$ is a power series with integer coefficients.)

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_{n}, \mathscr{P}(\mathfrak{S}_{n}))$ equipped with the uniform probability. We define a random variable $Z_{n}$ by $Z_{n}(\sigma) = \nu(\sigma)$.
Calculate, for any natural integer $k \leqslant n$, $\lim_{n \rightarrow +\infty} \mathbb{P}\left(Z_{n} = k\right)$.

For $n$ a natural integer greater than or equal to 2, we consider the probability space $(\mathfrak{S}_n, \mathscr{P}(\mathfrak{S}_n))$ equipped with the uniform probability. We define a random variable $Z_n$ by $Z_n(\sigma) = \nu(\sigma)$.
Calculate, for any natural integer $k \leqslant n$, $\lim_{n \rightarrow +\infty} \mathbb{P}(Z_n = k)$.

Recall the hypothesis $b_1 e^{a_1} + \cdots + b_r e^{a_r} = 0$ of Proposition 1. Define rational numbers $s_1, \ldots, s_r$ by the formula $$(T - a_1) \cdots (T - a_r) = T^r - s_1 T^{r-1} - \cdots - s_{r-1} T - s_r.$$ Show the equality: for all $n \geq 0$, $$u_{n+r} = s_1 u_{n+r-1} + \cdots + s_r u_n.$$

We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
Prove that, for all $n$ in $\mathbb { N } ^ { * } , \left| b _ { n } \right| \leqslant 1$. Deduce an inequality on the radius of convergence of the power series $\sum _ { k \geqslant 0 } b _ { k } t ^ { k }$.

Show that there exists a real number $c_1 > 0$ such that: for all $n \geq 0$, $$|v_n| \leq c_1 \frac{A^{n+1}}{n+1}.$$

For all integers $n, k \geq 0$, define the rational number $v_n(k)$ as the coefficient of degree $n$ in the power series $$\left(1 - s_1 x - \cdots - s_r x^r\right)^k v(x) = \sum_{n=0}^{\infty} v_n(k) x^n.$$ Show that $v(x)$ is the power series expansion of a rational fraction if and only if there exists an integer $k \geq 0$ such that $\sum_{n=0}^{\infty} v_n(k) x^n$ is a polynomial.

For all integers $n, k \geq 0$, define the rational number $v_n(k)$ as the coefficient of degree $n$ in the power series $$\left(1 - s_1 x - \cdots - s_r x^r\right)^k v(x) = \sum_{n=0}^{\infty} v_n(k) x^n.$$ Observe the equality: for all $n \geq r$ and $k \geq 0$, $$v_n(k+1) = v_n(k) - s_1 v_{n-1}(k) - \cdots - s_r v_{n-r}(k).$$

We denote by $\mathcal{E}$ the set of functions $f : \mathbb{C} \rightarrow \mathbb{C}$ expandable as a power series with radius of convergence infinity. Using the functions $Q_n \in \mathcal{E}$ satisfying $Q_n(z+1) - Q_n(z) = nz^{n-1}$ for all $n \in \mathbb{N}^*$ and $z \in \mathbb{C}$, and the bound $|Q_n(z)| \leqslant a\,\mathrm{e}^{bn|z|}$ for constants $a,b \in \mathbb{R}_+^*$, deduce the existence of a solution in $\mathcal{E}$ to the equation $(E_h)$: $$\forall z \in \mathbb{C},\, f(z+1) - f(z) = h(z)$$ when $h \in \mathcal{E}$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that if $f$ is a function $E$, then the numerical series $\sum_{n=0}^{\infty} \frac{b_n}{n!} \alpha^n$ converges for every real number $\alpha$. We denote its value by $f(\alpha)$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f$ be a function $E$ that is not a polynomial. Show that there exists $R > 0$ such that the numerical series $\sum_{n=0}^{\infty} b_n \alpha^n$ diverges for every real number $\alpha$ with $|\alpha| > R$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Recall that $\widehat{f}(x)$ denotes the Laplace transform $\widehat{f}(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} b_n x^n$. Which functions $E$ are such that $\widehat{f}$ is also a function $E$?

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Prove that functions $E$ are closed under addition and multiplication.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f$ be an exponential polynomial (i.e., $f(x) = \sum_{i=1}^{s} P_i(x) e^{c_i x}$ with $c_i \in \mathbf{Q}$ and $P_i \in \mathbf{Q}[x]$). Show that $f$ is a function $E$ such that $\widehat{f}$ is the power series expansion of a rational fraction with rational poles.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that if $\sum_{n=0}^{\infty} b_n x^n$ is the power series expansion of a rational fraction with rational poles, then $\sum_{n=0}^{\infty} \frac{b_n}{n!} x^n$ is a function $E$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that the Bessel function $$J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$$ is a function $E$ such that $\widehat{J}_0(x)$ satisfies the equation $(1 + x^2)\widehat{J}_0(x)^2 = 1$. Deduce that $J_0(x)$ is not an exponential polynomial.

Show that the real zeros of the Bessel function $J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$ are simple, that is, if $J_0(\alpha) = 0$, then $J_0'(\alpha) \neq 0$.

A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f(x)$ be a function $E$ such that $f(1) = 0$. Show that the power series $f(x)/(x-1)$ is still a function $E$.

Show that every Dirichlet series $\sum _ { n \geq 0 } f _ { n }$ converges uniformly on $\mathbf { R } _ { + }$. We then denote its sum by $f$. Justify that $f$ is continuous on $\mathbf { R } _ { + }$.
A Dirichlet series satisfies $f_n(x) = a_n e^{-\lambda_n x}$ with $\left| a _ { n } \right| \leq \frac { M } { 2 ^ { n } }$, $\lambda_0 = 0$, $\lim_{n\to+\infty}\lambda_n = +\infty$, and $\lambda_n = O(n)$.