For $p \in \mathbb { N } ^ { * }$, let $h(x) = \mathrm{e}^{-x} P(x)$ where $P$ is a polynomial solution of $(E_p)$, with power series coefficients satisfying $b _ { n } = \frac { ( - 1 ) ^ { n - 1 } ( n + p - 1 ) ! } { p ! n ! ( n - 1 ) ! } b _ { 1 }$ for all $n \in \mathbb{N}^*$. We set $g _ { p } ( x ) = x ^ { p - 1 } \mathrm { e } ^ { - x }$. Justify that $g _ { p } ^ { ( p ) }$ is developable as a power series and deduce from Question 34 that, for all $x \in \mathbb { R }$, $$P ( x ) = C x \mathrm { e } ^ { x } g _ { p } ^ { ( p ) } ( x )$$ where $C$ is a real constant whose expression in terms of $b _ { 1 }$ and $p$ we will specify.

Let $t \in \mathbf{R}$ and $(i,j) \in \llbracket 1;N \rrbracket^2$, justify that the series $\sum_{n \geq 0} \frac{t^n K^n[i,j]}{n!}$ converges. We denote by $H_t \in \mathscr{M}_N(\mathbf{R})$ the matrix defined by $$\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$$

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Prove that $f$ is expandable as a power series on $]-1, 1[$.

Let $\varphi_{0}$ be the function defined on $\mathbb{R}$ by
$$\left\{ \begin{array}{l} \varphi_{0}(x) = e^{-1/x^{2}} \text{ if } x \neq 0 \\ \varphi_{0}(0) = 0 \end{array} \right.$$
a. Show that for all $n \in \mathbb{N}$ there exists a polynomial $P_{n}$ such that for $x \neq 0$ we have
$$\varphi_{0}^{(n)}(x) = P_{n}\left(\frac{1}{x}\right) e^{-1/x^{2}}$$
b. Show that $\varphi_{0}$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$ with compact support. For all $n \in \mathbb{N}$, we set $$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$ In this part we assume that $f$ is not identically zero.
Deduce that for all $B > 0$ we have
$$\frac{M_{n}}{B^{n} n!} \underset{n \rightarrow \infty}{\longrightarrow} +\infty$$

We denote $\omega(t) = e^{2i\pi t}$ for $t \in [0,1]$. For all $p \in \mathbb{Z}$, we set $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Verify that this integral is well defined for all $p \in \mathbb{Z}$.

Let $\psi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ such that, for all $x \in \mathbb{R}$, $$\psi(x) = \begin{cases} \dfrac{x}{\mathrm{e}^{x} - 1} & \text{if } x \neq 0 \\ 1 & \text{otherwise} \end{cases}$$ Let furthermore $u$ be the function from $\mathbb{R}^{2}$ to $\mathbb{R}$ such that, for all $(x,t) \in \mathbb{R}^{2}$, $$u(x,t) = \psi(x)\,\mathrm{e}^{tx}.$$ Show that $u$ is of class $\mathcal{C}^{\infty}$ on $\mathbb{R}^{2}$.

Let $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ have slow growth and $t \in \mathbf{R}_{+}$.
Show that $x \in \mathbb{R} \mapsto P_{t}(f)(x)$ is of class $C^{2}$ on $\mathbf{R}$. Also show that
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime}(x) = \mathrm{e}^{-t} \int_{-\infty}^{+\infty} f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$
and
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime\prime}(x) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} f^{\prime\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$$

Let $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth and $t \in \mathbf{R}_+$. Show that $x \in \mathbf{R} \mapsto P_t(f)(x)$ is of class $C^2$ on $\mathbf{R}$. Also show that $$\forall x \in \mathbf{R}, \quad P_t(f)'(x) = \mathrm{e}^{-t}\int_{-\infty}^{+\infty} f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y$$ and $$\forall x \in \mathbf{R}, \quad P_t(f)''(x) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} f''\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

Show that $v(x)$ is not the power series expansion of a rational fraction.