We define a sequence of polynomials $\left( F _ { i } \right) _ { i \geqslant 0 }$ by the following recurrence formula: $$\begin{aligned} F _ { 0 } & = f _ { 0 } + f _ { 1 } X + \cdots + f _ { d } X ^ { d } \\ \text { for } i \geqslant 0 , \quad F _ { i + 1 } & = F _ { i } + R _ { i } \end{aligned}$$ where $R _ { i }$ denotes the remainder of the Euclidean division of $P$ by $F _ { i }$. We denote by $Q _ { i }$ the quotient of the Euclidean division of $P$ by $F _ { i }$. We set, for $i \in \mathbb { N }$: $$\alpha _ { i } = s ^ { - d } \cdot \left\| F _ { i } - X ^ { d } \right\| _ { r , s } ; \quad \beta _ { i } = \left\| 1 - Q _ { i } \right\| _ { r , s } ; \quad \varepsilon _ { i } = s ^ { - d } \cdot \left\| R _ { i } \right\| _ { r , s } .$$
Deduce that, for all $i \in \mathbb { N }$, we have:

• $\alpha _ { i } \leqslant \alpha _ { 0 } + 2 \cdot \left( 1 - 2 ^ { - i } \right) \cdot \varepsilon _ { 0 }$,
• $\beta _ { i } \leqslant \beta _ { 0 } + \left( 1 - 2 ^ { - i } \right) \cdot \varepsilon _ { 0 }$,
• $\varepsilon _ { i } \leqslant 2 ^ { - i } \cdot \varepsilon _ { 0 }$.

By continuing to bound the right-hand side of the equality in question 13, establish the estimate $$d_{VT}\left(p_{X_n}, \pi_1\right) \underset{n \rightarrow +\infty}{=} O\left(\frac{2^n}{(n+1)!}\right)$$ One may use binomial coefficients.

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. We call $\tilde{f}$ the application from $\mathbf{R}_+$ to $\mathbf{R}$, defined by: $$\forall x \in \mathbf{R}^+, \tilde{f}(x) = \ln(f(2x))$$
Show that $$\forall p \in \mathbf{N}^*, \forall x \in \mathbf{R}_+, \tilde{f}(x+p) = \tilde{f}(x) + \sum_{k=0}^{p-1} \ln\left(\frac{2x+2k+1}{2x+2k+2}\right)$$

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. We call $\tilde{f}$ the application from $\mathbf{R}_+$ to $\mathbf{R}$, defined by: $$\forall x \in \mathbf{R}^+, \tilde{f}(x) = \ln(f(2x))$$
Suppose here that $x \in \mathbf{R}_+^*$, $(n,p) \in (\mathbf{N}^*)^2$ and $x \leq p$. Verify that $$\tilde{f}(n) - \tilde{f}(n-1) \leq \frac{\tilde{f}(n+x) - \tilde{f}(n)}{x} \leq \frac{\tilde{f}(n+p) - \tilde{f}(n)}{p}$$ and that $(\tilde{f}(n+x) - \tilde{f}(n))$ has a limit as $n$ tends to $+\infty$.

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$. 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 $J$. The functional equation referred to as (1) is: $$\forall x \in I, (x+1)f(x) = (x+2)f(x+2)$$
Conclude that $f$ is the unique application from $I$ to $\mathbf{R}$, which is log-convex, which satisfies (1) and such that $$f(0) = \frac{\pi}{2}$$

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
Show that there exists a unique shift-invariant and invertible endomorphism $U$ such that $T = D \circ U$. Specify $U$ in the case $T = D$, then in the case $T = L$.

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 $$\forall t \in ]-T, +\infty[, (t+T)g(t) = (t+2T)g(t+2T).$$

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
For every polynomial $p \in \mathbb{K}[X]$ non-zero, verify that $\deg(Tp) = \deg(p) - 1$. Deduce $\ker(T)$ and the spectrum of $T$.

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 $J$.
Does there exist an application $h$, from $\mathbf{R}$ to $\mathbf{R}$ and log-convex, satisfying $$\forall t \in \mathbf{R}, (t+T)h(t) = (t+2T)h(t+2T)?$$

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. We introduce the set $I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$ and assume that $n$ and $k$ vary such that $k \in I _ { n }$.
Show that we have $$k ! ( n - k ) ! = 2 \pi \mathrm { e } ^ { - n } k ^ { k + 1 / 2 } ( n - k ) ^ { n - k + 1 / 2 } \left( 1 + O \left( \frac { 1 } { n } \right) \right)$$ as $n$ tends to infinity. One may use Stirling's formula: $n ! = \left( \frac { n } { \mathrm { e } } \right) ^ { n } \sqrt { 2 \pi n } \left( 1 + O \left( \frac { 1 } { n } \right) \right)$.

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 $B _ { n }$ is defined as in Q19. We assume $k \in I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$.
Deduce that, as $n$ tends to $+ \infty$, we have $$B _ { n } \left( x _ { n , k } \right) = \frac { 1 } { \sqrt { 2 \pi } } \frac { 1 + O \left( \frac { 1 } { n } \right) } { \left( \frac { 2 k } { n } \right) ^ { k + 1 / 2 } \left( 2 - \frac { 2 k } { n } \right) ^ { n - k + 1 / 2 } }$$

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, let $T_n$ denote the restriction of $T$ to $\mathbb{K}_n[X]$.
Determine $\operatorname{Im}(T_n)$ in terms of $n \in \mathbb{N}$ and deduce that $T$ is surjective.

We wish to show that, for every delta endomorphism $Q$, there exists a unique sequence of polynomials $(q_n)_{n \in \mathbb{N}}$ of $\mathbb{K}[X]$ such that:

• $q_0 = 1$;
• $\forall n \in \mathbb{N}, \deg(q_n) = n$;
• $\forall n \in \mathbb{N}^*, q_n(0) = 0$;
• $\forall n \in \mathbb{N}^*, Q q_n = q_{n-1}$.

Let $Q$ be a delta endomorphism. Show the existence and uniqueness of the sequence $(q_n)_{n \in \mathbb{N}}$ of polynomials associated with $Q$.

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Show that, for every natural number $n$, $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$

Let $(q_n)_{n \in \mathbb{N}}$ be a sequence of polynomials of $\mathbb{K}[X]$ such that $\forall n \in \mathbb{N}, \deg(q_n) = n$ and $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$
Show that there exists a unique delta endomorphism $Q$ for which $(q_n)_{n \in \mathbb{N}}$ is the associated sequence of polynomials.

Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number.
Show that the family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.

For $Q = D$, verify that $$\forall n \in \mathbb{N}, \quad q_n = \frac{X^n}{n!}$$

For $Q = E_1 - I$, verify that $$\forall n \in \mathbb{N}^*, \quad q_n = \frac{X(X-1)\cdots(X-n+1)}{n!}$$

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Prove that, for all $p \in \mathbb{K}[X]$, the expression $\sum_{k=0}^{+\infty} (Q^k p)(0) q_k$ makes sense and defines a polynomial of $\mathbb{K}[X]$, then that $$p = \sum_{k=0}^{+\infty} (Q^k p)(0) q_k$$

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 $$T = \sum_{k=0}^{+\infty} (T q_k)(0) Q^k$$

By choosing $Q = E_1 - I$, prove that, if $p$ is a non-constant polynomial, then $$p'(X) = \sum_{k=1}^{\deg(p)} \frac{1}{k} \left( \sum_{j=0}^k (-1)^{j+1} \binom{k}{j} p(X+j) \right)$$ This is the formula for numerical differentiation of polynomials.

If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
Show that, if there exists $(a_n)_{n \in \mathbb{N}}$ a sequence of scalars such that $T = \sum_{k=0}^{+\infty} a_k D^k$, then $T' = \sum_{k=1}^{+\infty} k a_k D^{k-1}$.

If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
If $T$ is a shift-invariant endomorphism, show that $T'$ is still a shift-invariant endomorphism.

If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
If $T$ is a delta endomorphism, show that $T'$ is a shift-invariant and invertible endomorphism.

Let $S$ and $T$ be two endomorphisms of $\mathbb{K}[X]$. The Pincherle derivative of an endomorphism $T$ is defined by $T'(p) = T(Xp) - XT(p)$.
Verify that $(S \circ T)' = S' \circ T + S \circ T'$.