In the three following cases, determine $E$.
II.B.1) $f(t) = \lambda^{\prime}(t)$, with $\lambda$ assumed to be of class $C^1$.
II.B.2) $f(t) = e^{t\lambda(t)}$.
II.B.3) $f(t) = \dfrac{e^{-t\lambda(t)}}{1+t^2}$.

In this question, we study the case $\lambda(t) = t^2$ and $f(t) = \dfrac{1}{1+t^2}$ for all $t \in \mathbb{R}^+$.
II.C.1) Determine $E$. What is the value of $Lf(0)$?
II.C.2) Prove that $Lf$ is differentiable.
II.C.3) Show the existence of a constant $A > 0$ such that for all $x > 0$, we have $$Lf(x) - (Lf)^{\prime}(x) = \frac{A}{\sqrt{x}}.$$
II.C.4) We denote $g(x) = e^{-x}Lf(x)$ for $x \geqslant 0$.
Show that for all $x \geqslant 0$, we have $$g(x) = \frac{\pi}{2} - A\int_0^x \frac{e^{-t}}{\sqrt{t}}\,dt.$$
II.C.5) Deduce from this the value of the integral $\displaystyle\int_0^{+\infty} e^{-t^2}\,dt$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$.
Show that $f$ extends by continuity at 0.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Determine $E$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Using a series expansion, show that for all $x > 0$, we have $$Lf(x) = \frac{1}{2x^2} - \frac{1}{x} + \sum_{n=1}^{+\infty} \frac{1}{(n+x)^2}.$$

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{t}{e^t - 1} - 1 + \dfrac{t}{2}$ for all $t \in \mathbb{R}^{+*}$ (extended by continuity at 0).
Does $Lf(x) - \dfrac{1}{2x^2} + \dfrac{1}{x}$ admit a finite limit at $0^+$?

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that if $E$ is not empty and if $\alpha$ is its infimum (we agree that $\alpha = -\infty$ if $E = \mathbb{R}$), then $Lf$ is of class $C^{\infty}$ on $]\alpha, +\infty[$ and express its successive derivatives using an integral.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
In the particular case where $f(t) = e^{-at}t^n$ for all $t \in \mathbb{R}^+$, with $n \in \mathbb{N}$ and $a \in \mathbb{R}$, make explicit $E$, $E^{\prime}$ and calculate $Lf(x)$ for $x \in E^{\prime}$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $E$ is not empty and that $f$ admits near 0 the following limited expansion of order $n \in \mathbb{N}$: $$f(t) = \sum_{k=0}^{n} \frac{a_k}{k!}t^k + O\left(t^{n+1}\right).$$
IV.C.1) Show that for all $\beta > 0$, we have, as $x$ tends to $+\infty$, the following asymptotic expansion: $$\int_0^{\beta} \left(f(t) - \sum_{k=0}^{n} \frac{a_k}{k!}t^k\right)e^{-tx}\,dt = O\left(x^{-n-2}\right).$$
IV.C.2) Deduce from this that as $x$ tends to infinity, we have the asymptotic expansion: $$Lf(x) = \sum_{k=0}^{n} \frac{a_k}{x^{k+1}} + O\left(x^{-n-2}\right).$$

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$. We assume that $f$ admits a finite limit $l$ at $+\infty$.
IV.D.1) Show that $E$ contains $\mathbb{R}^{+*}$.
IV.D.2) Show that $xLf(x)$ tends to $l$ at $0^+$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E$ does not contain 0.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E = ]0, +\infty[$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Show that $E^{\prime}$ contains 0.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
Calculate $(Lf)^{\prime}(x)$ for $x \in E$.

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
We denote for $n \in \mathbb{N}$ and $x \geqslant 0$, $$f_n(x) = \int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t} e^{-xt}\,dt.$$ Show that $\sum_{n \geqslant 0} f_n$ converges uniformly on $[0, +\infty[$.

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients and we use the transformation $L$ applied to elements of $\mathcal{P}$ for the study of an operator $U$.
Let $P$ and $Q$ be two elements of $\mathcal{P}$.
Show that the integral $\displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, where $\bar{P}$ is the polynomial whose coefficients are the conjugates of those of $P$, converges.

Let $x$ be a linear recurrent sequence. Show that the set $J_x$ of polynomials $A$ such that $A(\sigma)(x) = 0$ is an ideal of $\mathbb{K}[X]$, not reduced to $\{0\}$.
We recall that this implies two things:

• on the one hand, there exists in $J_x$ a unique monic polynomial $B$ of minimal degree;
• on the other hand, the elements of $J_x$ are the multiples of $B$.

By definition, we say that $B$ is the minimal polynomial of the sequence $x$, that the degree of $B$ is the minimal order of $x$, and that the relation $B(\sigma)(x) = 0$ is the minimal recurrence relation of $x$.

In $\mathbb{K}^{\mathbb{N}}$, what are the linear recurrent sequences of order 0? of order 1?
What are the sequences in $\mathbb{K}^{\mathbb{N}}$ whose minimal polynomial is $(X-1)^2$?

We consider the sequence $x$ defined by $x_0 = 0, x_1 = -1, x_2 = 2$ and by the linear recurrence relation of order 3: $\forall n \in \mathbb{N}, x_{n+3} = -3x_{n+2} - 3x_{n+1} - x_n$.
Determine the minimal polynomial (and thus the minimal order) of the sequence $x$.

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Prove that $\mathcal{R}_A(\mathbb{K})$ is a vector subspace of dimension $p$ of $\mathbb{K}^{\mathbb{N}}$ and that it is stable under $\sigma$ (we do not ask here to determine a basis of $\mathcal{R}_A(\mathbb{K})$, as this is the object of the following questions).

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic.
Determine $\mathcal{R}_A(\mathbb{K})$ when $A = X^p$ (with $p \geqslant 1$) and give a basis for it.

Let $A = \sum_{k=0}^{p} a_k X^k$ be an element of $\mathbb{K}[X]$, of degree $p \geqslant 0$, which without loss of generality we assume to be monic. In this question, we assume $p \geqslant 1$ and $A = (X - \lambda)^p$, with $\lambda$ in $\mathbb{K}^*$.
We denote by $E_A(\mathbb{K})$ the set of $x$ in $\mathbb{K}^{\mathbb{N}}$ with general term $x_n = Q(n)\lambda^n$, where $Q$ is in $\mathbb{K}_{p-1}[X]$.
a) Show that $E_A(\mathbb{K})$ is a vector subspace of $\mathbb{K}^{\mathbb{N}}$ and specify its dimension.
b) Show the equality $\mathcal{R}_A(\mathbb{K}) = E_A(\mathbb{K})$.

In this question, we assume that the polynomial $A$ is split over $\mathbb{K}$. More precisely, we write $A = X^{m_0} \prod_{k=1}^{d} (X - \lambda_k)^{m_k}$, where:

• the scalars $\lambda_1, \lambda_2, \ldots, \lambda_d$ are the distinct non-zero roots of $A$ in $\mathbb{K}$, and $m_1, m_2, \ldots, m_d$ are their respective multiplicities (greater than or equal to 1). If $A$ has no non-zero root, we adopt the convention that $d = 0$ and that $\prod_{k=1}^{d}(X-\lambda_k)^{m_k} = 1$;
• the integer $m_0$ is the multiplicity of 0 as a possible root of $A$. If 0 is not a root of $A$, we adopt the convention $m_0 = 0$.

With these notations, we have $\sum_{k=0}^{d} m_k = \deg A = p$.
Using the kernel decomposition theorem, show that $\mathcal{R}_A(\mathbb{K})$ is the set of sequences $x = (x_n)_{n \geqslant 0}$ in $\mathbb{K}^{\mathbb{N}}$ such that: $$\forall n \geqslant m_0, \quad x_n = \sum_{k=1}^{d} Q_k(n) \lambda_k^n$$ where, for all $k$ in $\{1, \ldots, d\}$, $Q_k$ is in $\mathbb{K}[X]$ with $\deg Q_k < m_k$.
Remark: if $d = 0$, the sum $\sum_{k=1}^{d} Q_k(n)\lambda_k^n$ is by convention equal to 0.

Let $x$ be a non-zero linear recurrent sequence, of order $m \geqslant 1$. Let $p = \operatorname{rang}(H_m(x))$. The kernel of $H_{p+1}(x)$ is a one-dimensional vector space whose a direction vector can be written $(b_0, \ldots, b_{p-1}, 1)$, where $b_0, \ldots, b_{p-1}$ are in $\mathbb{K}$.
With these notations, show that the minimal polynomial of $x$ is $B = X^p + b_{p-1}X^{p-1} + \cdots + b_1 X + b_0$.

We consider the sequence $x = (x_n)_{n \geqslant 0}$ defined by $$x_0 = 1, \quad x_1 = 1, \quad x_2 = 1, \quad x_3 = 0, \quad \text{and} \quad \forall n \in \mathbb{N}, x_{n+4} = x_{n+3} - 2x_{n+1}$$
In the computer language of your choice (which you will specify), write a procedure (or function) with parameter a natural number $n$ and returning the list (or sequence, or vector) of $x_k$ for $0 \leqslant k \leqslant n$.