In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. We have established that $q(x_0) < \alpha \Rightarrow \forall t \geqslant 0,\ q(f_{x_0})(t) \leqslant e^{-\beta t} q(x_0)$.
Deduce the existence of three strictly positive constants $\tilde { \alpha } , C$ and $\beta$ such that: $$\forall x _ { 0 } \in B ( 0 , \tilde { \alpha } ) , \quad \forall t \in \mathbf { R } _ { + } , \quad \left\| f _ { x _ { 0 } } ( t ) \right\| \leqslant C e ^ { - \beta t } \left\| x _ { 0 } \right\| ,$$ where $B ( 0 , \tilde { \alpha } )$ denotes the open ball, for the norm $\|.\|$, with center 0 and radius $\tilde { \alpha }$.

We seek to show that if $(M_{n})_{n \geqslant 0}$ is a sequence of strictly positive real numbers such that the series $\sum_{n \geqslant 1} \frac{M_{n-1}}{M_{n}}$ converges, there exists a function $f \in \mathcal{C}_{c}^{\infty}(\mathbb{R})$ not identically zero such that for all $n \geqslant 0$, $\|f^{(n)}\|_{\infty} \leqslant M_{n}$.
Conclude regarding the initially posed question.

An application $h$ from a non-trivial interval $J$ of $\mathbf{R}$ to $\mathbf{R}$ is said to be log-convex if, and only if, it takes values in $\mathbf{R}_+^*$ and $\ln \circ h$ is convex on $J$.
More generally, determine, if $T \in \mathbf{R}_+^*$, all applications $g$ from $]-T, +\infty[$ to $\mathbf{R}$, log-convex and satisfying $$\forall t \in ]-T, +\infty[, (t+T)g(t) = (t+2T)g(t+2T).$$

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0.$$
For all $n \in \mathbb { N } ^ { * }$, show that $B _ { n }$ is a decreasing function on $\mathbb { R } ^ { + }$. One may distinguish according to whether $n$ is even or odd.

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Let $x, y \in A$ such that $xy = yx$ and such that $V = \mathbb{R}x + \mathbb{R}y$ is of dimension 2 over $\mathbb{R}$. Show that $$\forall u, v \in V \quad \|u+v\|^2 + \|u-v\|^2 \geq 4\|u\| \cdot \|v\|$$ and that the restriction of $\|\cdot\|$ to $V$ comes from an inner product on $V$.

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Conclude that $A$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$ (Theorem C).

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Deduce that, for every shift-invariant endomorphism $T$, we have $$T = \sum_{k=0}^{+\infty} (T q_k)(0) Q^k$$

Let $Q$ be a delta endomorphism. There exists a unique shift-invariant and invertible endomorphism $U$ such that $Q = D \circ U$. We denote by $(q_n)_{n \in \mathbb{N}}$ the sequence of polynomials associated with $Q$.
Prove that, for all $n \in \mathbb{N}^*$, we have $$\left(Q' \circ U^{-n-1}\right)\left(X^n\right) = X U^{-n}\left(X^{n-1}\right)$$

Let $Q$ be a delta endomorphism with $Q = D \circ U$ where $U$ is the unique shift-invariant and invertible endomorphism. We denote by $(q_n)_{n \in \mathbb{N}}$ the sequence of polynomials associated with $Q$.
Deduce that, for all $n \in \mathbb{N}^*$, $$n! q_n(X) = X U^{-n}\left(X^{n-1}\right)$$ then that $$n q_n(X) = X (Q')^{-1}\left(q_{n-1}\right)$$

We apply the results of question 40 to the endomorphism $L$ defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We denote by $(\ell_n)_{n \in \mathbb{N}}$ its associated sequence of polynomials.
Verify that, for $n \in \mathbb{N}^*$, $$\ell_n' = \ell_{n-1}' - \ell_{n-1}$$ and $$X\ell_n'' - X\ell_n' + n\ell_n = 0$$ and $$\ell_n(X) = \sum_{k=1}^n (-1)^k \binom{n-1}{k-1} \frac{X^k}{k!}$$

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Show that there exists a unique invertible endomorphism $T$ such that $$\forall n \in \mathbb{N}, \quad T q_n = \frac{X^n}{n!}$$

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$, and let $T$ be the unique invertible endomorphism such that $T q_n = \frac{X^n}{n!}$ for all $n \in \mathbb{N}$.
Also show that $D = T \circ Q \circ T^{-1}$.

We fix $\alpha > 0$ and define the function $W$ from $\mathbb{K}[X]$ by $$W : \begin{array}{ccc} \mathbb{K}[X] & \rightarrow & \mathbb{K}[X] \\ p & \mapsto & p(\alpha X) \end{array}$$
Show that $W$ is an automorphism of $\mathbb{K}[X]$.

We fix $\alpha > 0$ and define $W : p \mapsto p(\alpha X)$. We set $P = W \circ L \circ W^{-1}$ where $L$ is the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.
Show that $$P = \frac{1}{\alpha} D \circ \left(\frac{1}{\alpha} D - I\right)^{-1}$$

We fix $\alpha > 0$, define $W : p \mapsto p(\alpha X)$, and set $P = W \circ L \circ W^{-1}$ where $L$ is the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We have $P = \frac{1}{\alpha} D \circ \left(\frac{1}{\alpha} D - I\right)^{-1}$.
Show that $P$ is a delta endomorphism whose associated polynomial sequence $(p_n)_{n \in \mathbb{N}}$ satisfies $$\forall n \in \mathbb{N}, \quad p_n = \ell_n(\alpha X)$$

Let $L$ be the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$, and let $P = W \circ L \circ W^{-1}$ with $W : p \mapsto p(\alpha X)$.
Verify that $D = L \circ (L-I)^{-1}$ then that $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.

We denote by $T$ the unique automorphism satisfying, for all $n \in \mathbb{N}$, $T\ell_n = \frac{X^n}{n!}$ and we set $Q = T \circ P \circ T^{-1}$, where $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.
Show that $Q = D \circ (\alpha I + (1-\alpha)D)^{-1}$. Deduce that $Q$ is a delta endomorphism whose associated polynomial sequence $(r_n)_{n \in \mathbb{N}}$ satisfies $$\forall n \in \mathbb{N}^*, \quad r_n = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \frac{X^k}{k!}$$

Using the results of the previous questions, conclude that $$\forall n \in \mathbb{N}^*, \quad \ell_n(\alpha X) = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \ell_k(X)$$

Show that $\Delta$ is an endomorphism of $\mathbb{K}[X]$, where $$\Delta : \begin{cases} \mathbb{K}[X] \rightarrow \mathbb{K}[X] \\ P(X) \mapsto P(X+1) - P(X) \end{cases}$$

Show that any function bounded in absolute value by a polynomial function in $|x|$ has slow growth.

Let $f : [a, b] \longrightarrow \mathbf{R}$ be a continuous function. Prove that the restriction $g$ of the function $f$ to the interval $]a, b[$ belongs to the set $\mathscr{D}_{a,b}$.

Give a necessary and sufficient condition on $R_u$ for $\mathbb{M}_n(u) = \emptyset$ and give an example of $u$ for which this equality holds.

For a strictly positive rational number $a \in \mathbf{Q}_{>0}$, let $\log(a)$ denote the unique real number satisfying $e^{\log a} = a$. Deduce from Theorem 1 that $\log(a)$ is irrational for every strictly positive rational number $a \neq 1$.
(Theorem 1: Let $r \geq 2$ be an integer. If $a_1, \ldots, a_r \in \mathbf{Q}$ are distinct rational numbers, then the real numbers $e^{a_1}, \ldots, e^{a_r}$ are linearly independent over $\mathbf{Q}$.)

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_n(\mathbb{R})$ $$M_x = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right).$$ Deduce that for all $x \in \mathbb{R}$, we have $$\sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma) x^{\nu(\sigma)} = (x-1)^{n-1}(x+n-1).$$

Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$, where $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$ and $L^1(\varphi) = \{f \in C^0(\mathbf{R}),\, f\varphi \text{ integrable on } \mathbf{R}\}$.