We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ Deduce that: $$F_{n}(x) = G_{n}(x) - \int_{0}^{n+1} \frac{h(u)}{u+x} du$$ where $$G_{n}(x) = \ln n! + (x+1)\ln n - \left(x+n+\frac{3}{2}\right)\ln(x+n+1) + n+1 + \left(x+\frac{1}{2}\right)\ln x$$

We fix $x > 0$ and for all natural integers $n$, we define $F_{n}(x)$ by: $$F_{n}(x) = \ln\left(\frac{n! \, n^{x+1}}{(x+1)(x+2) \ldots (x+n+1)}\right)$$ and $$G_{n}(x) = \ln n! + (x+1)\ln n - \left(x+n+\frac{3}{2}\right)\ln(x+n+1) + n+1 + \left(x+\frac{1}{2}\right)\ln x$$
V.C.1) Using Stirling's formula, show that: $$\lim_{n \rightarrow +\infty} G_{n}(x) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi}$$
V.C.2) Deduce that: $$\ln \Gamma(x+1) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi} - \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$

Using the identity $$\ln \Gamma(x+1) = \left(x+\frac{1}{2}\right)\ln x - x + \ln\sqrt{2\pi} - \int_{0}^{+\infty} \frac{h(u)}{u+x} du$$ show that for all strictly positive real $x$, $$\frac{\Gamma^{\prime}(x+1)}{\Gamma(x+1)} = \ln x + \frac{1}{2x} + \int_{0}^{+\infty} \frac{h(u)}{(u+x)^{2}} du$$

Let $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$ be four strictly positive real numbers pairwise distinct and two strictly positive real numbers $E$ and $N$. Let $\Omega$ be the part, assumed to be non-empty, formed of the quadruplets $x = (x_{1}, x_{2}, x_{3}, x_{4})$ of $\mathbb{R}_{+}^{4}$ satisfying: $$\left\{\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} = N \\ \varepsilon_{1} x_{1} + \varepsilon_{2} x_{2} + \varepsilon_{3} x_{3} + \varepsilon_{4} x_{4} = E \end{array}\right.$$
VI.A.1) Let $f$ be a function of class $\mathcal{C}^{1}$ on $\mathbb{R}_{+}^{4}$. Show that $f$ admits a maximum on $\Omega$. We then denote $a = (a_{1}, a_{2}, a_{3}, a_{4}) \in \Omega$ a point at which this maximum is attained.
VI.A.2) Show that if $(x_{1}, x_{2}, x_{3}, x_{4}) \in \Omega$ then $x_{3}$ and $x_{4}$ can be written in the form $$\begin{aligned} & x_{3} = u x_{1} + v x_{2} + w \\ & x_{4} = u^{\prime} x_{1} + v^{\prime} x_{2} + w^{\prime} \end{aligned}$$ where we shall give explicitly $u, v, u^{\prime}, v^{\prime}$ in terms of $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4}$.
VI.A.3) Assuming that none of the numbers $a_{1}, a_{2}, a_{3}, a_{4}$ is zero, deduce that $$\begin{aligned} & \frac{\partial f}{\partial x_{1}}(a) + u \frac{\partial f}{\partial x_{3}}(a) + u^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \\ & \frac{\partial f}{\partial x_{2}}(a) + v \frac{\partial f}{\partial x_{3}}(a) + v^{\prime} \frac{\partial f}{\partial x_{4}}(a) = 0 \end{aligned}$$
VI.A.4) Show that the vector subspace of $\mathbb{R}^{4}$ spanned by the vectors $(1, 0, u, u^{\prime})$ and $(0, 1, v, v^{\prime})$ admits a supplementary orthogonal subspace spanned by the vectors $(1,1,1,1)$ and $(\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}, \varepsilon_{4})$.
VI.A.5) Deduce the existence of two real numbers $\alpha, \beta$ such that for all $i \in \{1,2,3,4\}$ we have $$\frac{\partial f}{\partial x_{i}}(a) = \alpha + \beta \varepsilon_{i}$$

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Determine a family of polynomials $\left(K_n\right)_{n \in \mathbb{N}}$ satisfying the following two conditions:
i. for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ and its leading coefficient is strictly positive;
ii. for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for the inner product $\langle \cdot, \cdot \rangle$.
Justify the uniqueness of such a family.

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm. For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Show that the sequence $\left(\left\|\Pi_n - f\right\|\right)_{n \in \mathbb{N}}$ is decreasing and converges to 0.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$. The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the orthonormal family defined in question II.E.
Express $s_n$ using the sequence of polynomials $\left(K_p\right)_{p \in \mathbb{N}}$.

The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the unique family of polynomials such that for all $p \in \mathbb{N}$, the degree of $K_p$ equals $p$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_p\right)_{0 \leqslant p \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
For all $p \in \llbracket 0; n-1 \rrbracket$, calculate $K_p(1)$.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Determine the value of $s_n$.

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $\epsilon \in ] 0 , \pi [$ and $k \in \mathbb { N } ^ { * }$. By writing $f _ { k } ( u , v ) - f ( u , v )$ as a sum of two terms and applying Question 10, prove that for every $( u , v ) \in \mathbb { R } ^ { 2 }$: $$\left| f _ { k } ( u , v ) - f ( u , v ) \right| \leq 2 \epsilon \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \| d _ { k } ( \epsilon )$$

We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Prove the Ergodic Theorem for the function $f$. (One may set $M = 2 \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \|$. For given $\epsilon > 0$, one may choose $k \in \mathbb { N } ^ { * }$ such that $d _ { k } ( \epsilon ) < \epsilon$. Next, one may apply Question 14 to $f _ { k }$ and consider $T _ { 0 } > 0$ such that for every $T \geq T _ { 0 }$: $$\left| \frac { 1 } { T } \int _ { 0 } ^ { T } f _ { k } \circ \theta ( t ) d t - ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f _ { k } \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \right| < \epsilon .)$$

Let $a , b \in ] 0,2 \pi [$ such that $a < b$. We denote by $\phi _ { a , b } : \mathbb { R } \rightarrow \mathbb { R }$ the continuous $2 \pi$-periodic function defined as follows. The function $\phi _ { a , b }$ is zero on $[ 0 , a ]$ and $[ b , 2 \pi ]$. For every $t \in [ a , b ] , \phi _ { a , b } ( t ) = \sin ^ { 2 } \left( \frac { \pi } { b - a } ( t - a ) \right)$.
Recall that every non-empty open set of $] - 1,1 [ ^ { 2 }$ contains a rectangle of the form $] \cos b , \cos a [ \times ] \cos d , \cos c [$ where $0 < a < b < \pi$ and $0 < c < d < \pi$.
Consider the solution $x ( t ) = \sum _ { i = 1 } ^ { 2 } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i }$ of (1) obtained by taking $c _ { 1 } = c _ { 2 } = 1$ in (2). Let $\Omega$ be a non-empty open set of $\left\{ u e _ { 1 } + v e _ { 2 } \mid u , v \in ] - 1,1 [ \right\}$. Prove that there exists $t \in [ 0 , + \infty [$ such that $x ( t ) \in \Omega$. (One may reason by contradiction and justify the existence of a function of the type $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \phi _ { a , b } \left( \theta _ { 1 } \right) \phi _ { c , d } \left( \theta _ { 2 } \right) = \Phi \left( \theta _ { 1 } , \theta _ { 2 } \right)$ such that $\Phi ( \theta ( t ) )$ is zero for all $t \in [ 0 , + \infty [)$.

In each of the two cases below, show that $f * g$ is defined and bounded on $\mathbb{R}$ and give an upper bound for $\|f * g\|_{\infty}$ which may involve $\|\cdot\|_{1}$, $\|\cdot\|_{2}$ or $\|\cdot\|_{\infty}$. a) $f \in L^{1}(\mathbb{R}),\ g \in C_{b}(\mathbb{R})$; b) $f, g \in L^{2}(\mathbb{R})$.

Let $f, g \in C(\mathbb{R})$ be such that $f * g(x)$ is defined for every real $x$. Show that $f * g = g * f$.

Show that if $f$ and $g$ have compact support, then $f * g$ has compact support.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Show that a function $h$ is uniformly continuous on $\mathbb{R}$ if and only if $\lim_{\alpha \rightarrow 0} \|T_{\alpha}(h) - h\|_{\infty} = 0$.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. For any real $\alpha$, show that $T_{\alpha}(f * g) = \left(T_{\alpha}(f)\right) * g$.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. For any real $\alpha$, show that $\left\|T_{\alpha}(f * g) - f * g\right\|_{\infty} \leqslant \left\|T_{\alpha}(f) - f\right\|_{2} \times \|g\|_{2}$.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Deduce that $f * g$ is uniformly continuous on $\mathbb{R}$ in the case where $f$ has compact support.

For any function $h$ in $C(\mathbb{R})$ and any real $\alpha$, we define the function $T_{\alpha}(h)$ by setting $T_{\alpha}(h)(x) = h(x - \alpha)$ for all $x \in \mathbb{R}$. We assume that $f$ and $g$ belong to $L^{2}(\mathbb{R})$. Show that $f * g$ is uniformly continuous on $\mathbb{R}$ in the general case.

Assume that $f \in L^{1}(\mathbb{R})$ and $g \in C_{b}(\mathbb{R})$. a) Show that $f * g$ is continuous. b) Show that if $g$ is uniformly continuous on $\mathbb{R}$, then $f * g$ is uniformly continuous on $\mathbb{R}$.

Let $k$ be a non-zero natural number. Assume that $g$ is of class $C^{k}$ on $\mathbb{R}$ and that all its derivative functions, up to order $k$, are bounded on $\mathbb{R}$. Show that $f * g$ is of class $C^{k}$ on $\mathbb{R}$ and specify its derivative function of order $k$.

In this question I.C.3, assume that $g$ is continuous, $2\pi$-periodic and of class $C^{1}$ piecewise. a) State without proof the theorem on Fourier series applicable to continuous, $2\pi$-periodic functions of class $C^{1}$ piecewise. b) Show that $f * g$ is $2\pi$-periodic and is the sum of its Fourier series. Specify the Fourier coefficients of $f * g$ using the Fourier coefficients of $g$ and integrals involving $f$.

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges pointwise to $f$ on $\mathbb{R}$.

Let $f \in C_{b}(\mathbb{R})$ and let $(\delta_{n})$ be a sequence of functions forming an approximate identity. Show that if $f$ has compact support, then the sequence $\left(f * \delta_{n}\right)_{n \in \mathbb{N}}$ converges uniformly to $f$ on $\mathbb{R}$.