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

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 $(S \circ T)' = S' \circ T + S \circ T'$.

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

Let $R \in \mathrm{O}_{d}(\mathbb{R})$. Verify that $\operatorname{det}(R) \in \{-1, +1\}$.

Let $P \in \mathbb{K}[X]$. Determine the degree of $\Delta(P)$ as a function of that of $P$, where $\Delta(P) = P(X+1) - P(X)$.

Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$.
We admit throughout the rest of the problem that $\int_{-\infty}^{+\infty} \varphi(t) \mathrm{d}t = 1$.

Calculate $$\sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma), \quad \sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma) \nu(\sigma) \quad \text{and} \quad \sum_{\sigma \in \mathfrak{S}_{n}} \frac{\varepsilon(\sigma)}{\nu(\sigma)+1}.$$

Verify that $(A, B) \mapsto \langle A, B \rangle$ is an inner product on the vector space $\mathscr{M}_{d}(\mathbb{R})$. We denote by $\|A\| = \sqrt{\langle A, A \rangle}$ the associated norm.

Show that, for all $d \in \mathbb{N}^{*}$, $\Delta$ induces an endomorphism on $\mathbb{K}_{d}[X]$, where $\Delta(P) = P(X+1) - P(X)$.

• [(a)] Show that for all $u, v \in \mathbb{R}^{d}$ and $A \in \mathscr{M}_{d}(\mathbb{R})$, we have $\langle u, Av \rangle_{\mathbb{R}^{d}} = \langle uv^{T}, A \rangle$.
• [(b)] Show that $\operatorname{tr}(AB) = \operatorname{tr}(BA)$ for $A, B \in \mathscr{M}_{d}(\mathbb{R})$.
• [(c)] Deduce that for all $A, B$ and $C$ in $\mathscr{M}_{d}(\mathbb{R})$ we have $$\langle A, BC \rangle = \langle B^{T} A, C \rangle = \langle AC^{T}, B \rangle.$$

Let $f \in \mathcal { C } ^ { 0 } ( [ 0 , + \infty [ )$ and $\ell \in \mathbb { R }$. Prove that $$\left( \lim _ { x \rightarrow + \infty } f ( x + 1 ) - f ( x ) = \ell \right) \Rightarrow \left( \lim _ { x \rightarrow + \infty } \frac { f ( x ) } { x } = \ell \right)$$

We denote by $\Delta_{d}$ the endomorphism of $\mathbb{K}_{d}[X]$ induced by $\Delta$, where $\Delta(P) = P(X+1) - P(X)$. Determine $\operatorname{Ker}(\Delta_{d})$ and $\operatorname{Im}(\Delta_{d})$ for all $d \in \mathbb{N}^{*}$.

Show that if $H$ is a Hadamard matrix of order $n$ greater than or equal to 4, then $n$ is a multiple of 4. One may begin by showing that we can assume the first row of $H$ is composed only of 1's and its second row is composed of $n/2$ coefficients equal to 1 then $n/2$ coefficients equal to $-1$.

Let $A \in \mathscr{M}_n(\mathbb{C})$. Show the equivalence of the following two assertions
(i) $A \in \mathbb{M}_n(v)$ for every sequence $v = (v_k)_{k \geqslant 0}$ of $\mathbb{C}$ satisfying $R_v > 0$.
(ii) $A$ is nilpotent (that is, there exists $k \in \mathbb{N}^*$ such that $A^k = 0_n$).

Let $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d})$ be a diagonal matrix with positive coefficients and let $R \in \mathrm{O}_{d}(\mathbb{R})$.

• [(a)] Show that for all $1 \leqslant i \leqslant d$, we have $|R_{ii}| \leqslant 1$ where $R_{ii}$ is the $i$-th diagonal coefficient of $R$.
• [(b)] Deduce that $\langle D, R \rangle \leqslant \operatorname{tr}(D)$.

We consider the application $\Delta$ defined by $\Delta(P) = P(X+1) - P(X)$. Deduce $\operatorname{Ker}(\Delta)$ and $\operatorname{Im}(\Delta)$. Apply the results obtained to the study of the equation $(E_{h})$: $$\forall x \in \mathbb{K},\, f(x+1) - f(x) = h(x)$$ in the case where $h$ is a polynomial function.

Show that for every integer $m \geqslant 0$, we have $$D_{u^{(m)}} = D_u$$

For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.

• [(a)] Verify that for all $a, b \in \mathbb{R}^{d}$ and $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $|\phi_{g}(a) - \phi_{g}(b)| = |a - b|$.
• [(b)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $\phi_{g} = \phi_{g^{\prime}}$ if and only if $g = g^{\prime}$.
• [(c)] Show that there exists a unique $e \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{e}$ is the identity map on $\mathbb{R}^{d}$, that is $\phi_{e}(x) = x$ for all $x \in \mathbb{R}^{d}$.

Suppose (for this question only) that $h$ is the function $x \mapsto x$. Determine a solution of $(E_{h})$: $$\forall x \in \mathbb{K},\, f(x+1) - f(x) = x$$ in $\mathbb{K}_{2}[X]$, then all polynomial solutions of the equation $(E_{h})$.

Let $v = (v_k)_{k \geqslant 0}$ be another sequence of complex numbers. Show that $$\mathbb{M}_n(u) \cap \mathbb{M}_n(v) \subset \mathbb{M}_n(u+v) \cap \mathbb{M}_n(u \star v)$$