We assume $\alpha = 1$ and use the notation $V_n(z) = U_{n+1}(z,-1)$, and $$\varphi_1(y) : t \mapsto \left(1-t^2\right)y''(t) - 3t\,y'(t)$$ By differentiating twice the function $t \mapsto (\sin t)\,V_n(\cos t) - \sin((n+1)t)$, show that for all $n \in \mathbb{N}$, $V_n$ is an eigenvector of $\varphi_1$.

For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real (when the expression makes sense): $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
Show that, for all $t \in \mathbb{R}$, the application $$\mathrm{N}_t : \left|\begin{array}{rll} D(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & \mathrm{N}(x,y,t) \end{array}\right.$$ is harmonic.
One may use question II.B.3.

For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real: $$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
In the rest of this part, the pair $(x,y)$ is fixed in $D(0,1)$.
Show that $t \mapsto \mathrm{N}(x,y,t)$ is defined and continuous on $[0, 2\pi]$.

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$\mathrm{N}_f(x,y) = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{N}(x,y,t) f(\cos t, \sin t)\, \mathrm{d}t$$ on $D(0,1)$, and $$u(x,y) = \begin{cases} \mathrm{N}_f(x,y) & \text{if } (x,y) \in D(0,1) \\ f(x,y) & \text{if } (x,y) \in C(0,1) \end{cases}$$ on $\bar{D}(0,1)$.
a) Show that $\mathrm{N}_f$ admits a second-order partial derivative $\partial_{11} \mathrm{N}_f$ with respect to $x$.
Similarly, one can show that $\mathrm{N}_f$ admits second-order partial derivatives with respect to all its variables, continuous on $D(0,1)$. This result is admitted for the rest.
Express, for all $(x,y) \in D(0,1)$, for all $(i,j) \in \{1,2\}^2$, $\partial_{ij} \mathrm{N}_f(x,y)$ in terms of $\partial_{ij} \mathrm{N}(x,y,t)$.
b) Deduce that $u$ is harmonic on $D(0,1)$.

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$\mathrm{N}_f(x,y) = \frac{1}{2\pi} \int_0^{2\pi} \mathrm{N}(x,y,t) f(\cos t, \sin t)\, \mathrm{d}t$$ on $D(0,1)$.
In this question, we fix $t_0 \in [0,2\pi]$, $(x,y) \in D(0,1)$ and $\varepsilon > 0$. Moreover, we denote, for all real $\delta > 0$: $$I_0^\delta = \left\{ t \in [0,2\pi] \mid \|(\cos t, \sin t) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta \right\}$$
a) Show that $I_0^\delta$ is an interval or the union of two disjoint intervals.
The use of a drawing will be appreciated; however, this drawing will not constitute a proof.
b) Show, using the application $f$, the existence of a real $\delta > 0$ such that $$\left| \int_{t \in I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$
c) Let $\delta > 0$ be arbitrary. Show that, if $t \in [0,2\pi] \backslash I_0^\delta$ and $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \delta/2$, then $$|\mathrm{N}(x,y,t)| \leqslant 4 \frac{1 - (x^2 + y^2)}{\delta^2}$$
d) Deduce from the previous question that, for $\delta > 0$ fixed, there exists $\eta > 0$ such that, if $\|(x,y) - (\cos t_0, \sin t_0)\|_2 \leqslant \eta$, then $$\left| \int_{t \in [0,2\pi] \backslash I_0^\delta} \mathrm{N}(x,y,t) \left( f(\cos t, \sin t) - f(\cos t_0, \sin t_0) \right) \mathrm{d}t \right| \leqslant \frac{\varepsilon}{2}$$

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$u(x,y) = \begin{cases} \mathrm{N}_f(x,y) & \text{if } (x,y) \in D(0,1) \\ f(x,y) & \text{if } (x,y) \in C(0,1) \end{cases}$$ on $\bar{D}(0,1)$.
Prove that $u$ is an application continuous at every point of $C(0,1)$. What can be concluded about the application $u$?

We assume that $f$ is the zero application on $C(0,1)$ and that $u$ is an element of $\mathcal{D}_f$. For all $n \in \mathbb{N}$, we define the application $$u_n : \begin{array}{rll} \bar{D}(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & u(x,y) + \dfrac{1}{n}(x^2 + y^2) \end{array}$$
Suppose that $u_n$ admits a local maximum at $(\tilde{x}, \tilde{y}) \in D(0,1)$.
a) By examining the behavior of the function $x \mapsto u_n(x, \tilde{y})$ show that, in this case, $\partial_{11} u_n(\tilde{x}, \tilde{y}) \leqslant 0$. Similarly, one can show that $\partial_{22} u_n(\tilde{x}, \tilde{y}) \leqslant 0$. Thus $\Delta u_n(\tilde{x}, \tilde{y}) \leqslant 0$. This result is admitted for the rest.
b) Deduce that $u_n$ does not admit a local maximum on $D(0,1)$.

We assume that $f$ is the zero application on $C(0,1)$ and that $u$ is an element of $\mathcal{D}_f$. For all $n \in \mathbb{N}$, we define the application $$u_n : \begin{array}{rll} \bar{D}(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & u(x,y) + \dfrac{1}{n}(x^2 + y^2) \end{array}$$
Deduce that, for all $(x,y) \in D(0,1)$, $u_n(x,y) \leqslant 1/n$.

We assume that $f$ is the zero application on $C(0,1)$ and that $u$ is an element of $\mathcal{D}_f$. For all $n \in \mathbb{N}$, we define the application $$u_n : \begin{array}{rll} \bar{D}(0,1) & \rightarrow & \mathbb{R} \\ (x,y) & \mapsto & u(x,y) + \dfrac{1}{n}(x^2 + y^2) \end{array}$$
Show that $u$ is identically zero on $\bar{D}(0,1)$.

We recall that the function $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Sketch the graph of $\phi$. Show that $\phi$ defines by restriction to the intervals $] - 1,0 [$ and $] 0 , + \infty [$ respectively

• a bijection $\left. \phi _ { - } : \right] - 1,0 [ \rightarrow ] 0 , + \infty [$,
• a bijection $\left. \phi _ { + } : \right] 0 , + \infty [ \rightarrow ] 0 , + \infty [$.

We denote $\left. \phi _ { - } ^ { - 1 } : \right] 0 , + \infty [ \rightarrow ] - 1,0 \left[ \right.$ and $\left. \phi _ { + } ^ { - 1 } : \right] 0 , + \infty [ \rightarrow ] 0 , + \infty [$ the inverse bijections.

What is the domain of definition $\mathcal{D}$ of the function $\Gamma$, where for $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$?

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. For all $x \in \mathcal{D}$, express $\Gamma(x+1)$ in terms of $x$ and $\Gamma(x)$.
Deduce from this, for all $x \in \mathcal{D}$ and all $n \in \mathbb{N}^{*}$, an expression for $\Gamma(x+n)$ in terms of $x$, $n$ and $\Gamma(x)$, as well as the value of $\Gamma(n)$ for all $n \geqslant 1$.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show the existence of the two integrals $\int_{0}^{+\infty} e^{-t^{2}} \mathrm{~d}t$ and $\int_{0}^{+\infty} e^{-t^{4}} \mathrm{~d}t$ and express them using $\Gamma$.

Let $a$ and $b$ be two real numbers such that $0 < a < b$. Show that, for all $t > 0$ and all $x \in [a, b]$,
$$t^{x} \leqslant \max\left(t^{a}, t^{b}\right) \leqslant t^{a} + t^{b}$$

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma$ is of class $\mathcal{C}^{\infty}$ on $\mathcal{D}$.
Let $k \in \mathbb{N}^{*}$ and $x \in \mathcal{D}$. Express $\Gamma^{(k)}(x)$, the $k$-th derivative of $\Gamma$ at point $x$, in the form of an integral.

For $x \in \mathbb{R}$, $\Gamma(x) = \int_{0}^{+\infty} t^{x-1} \mathrm{e}^{-t} \mathrm{~d}t$. Show that $\Gamma^{\prime}$ vanishes at a unique real number $\xi$ whose integer part will be determined.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ For a non-constant polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\delta(P))$ and $\operatorname{cd}(\delta(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ Deduce the kernel $\operatorname{ker}(\delta)$ and the image $\operatorname{Im}(\delta)$ of the endomorphism $\delta$.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ More generally, for $j \in \llbracket 1, n \rrbracket$, show the following equalities: $$\operatorname{ker}\left(\delta^j\right) = \mathbb{R}_{j-1}[X] \quad \text{and} \quad \operatorname{Im}\left(\delta^j\right) = \mathbb{R}_{n-j}[X]$$

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ For $k \in \mathbb{N}$ and $P \in \mathbb{R}_n[X]$, express $\delta^k(P)$ in terms of $\tau^j(P)$ for $j \in \llbracket 0, k \rrbracket$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Calculate $\delta\left(H_0\right)$ and, for $k \in \llbracket 1, n \rrbracket$, express $\delta\left(H_k\right)$ in terms of $H_{k-1}$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that, for $k, l \in \llbracket 0, n \rrbracket$, $$\delta^k\left(H_l\right)(0) = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$$

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that, for every $P \in \mathbb{R}_n[X]$, $$P = \sum_{k=0}^{n} \left(\delta^k(P)\right)(0) H_k$$

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Give the coordinates of the polynomial $X^3 + 2X^2 + 5X + 7$ in the basis $(H_0, H_1, H_2, H_3)$ of $\mathbb{R}_3[X]$.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce a polynomial $P \in \mathbb{R}_5[X]$ such that $$\delta^2(P) = X^3 + 2X^2 + 5X + 7$$