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

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

Let $Q$ be a delta endomorphism. There exists a unique shift-invariant and invertible endomorphism $U$ such that $Q = D \circ U$. We denote by $(q_n)_{n \in \mathbb{N}}$ the sequence of polynomials associated with $Q$.
Prove that, for all $n \in \mathbb{N}^*$, we have $$\left(Q' \circ U^{-n-1}\right)\left(X^n\right) = X U^{-n}\left(X^{n-1}\right)$$

Let $Q$ be a delta endomorphism with $Q = D \circ U$ where $U$ is the unique shift-invariant and invertible endomorphism. We denote by $(q_n)_{n \in \mathbb{N}}$ the sequence of polynomials associated with $Q$.
Deduce that, for all $n \in \mathbb{N}^*$, $$n! q_n(X) = X U^{-n}\left(X^{n-1}\right)$$ then that $$n q_n(X) = X (Q')^{-1}\left(q_{n-1}\right)$$

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Show that there exists a unique invertible endomorphism $T$ such that $$\forall n \in \mathbb{N}, \quad T q_n = \frac{X^n}{n!}$$

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$, and let $T$ be the unique invertible endomorphism such that $T q_n = \frac{X^n}{n!}$ for all $n \in \mathbb{N}$.
Also show that $D = T \circ Q \circ T^{-1}$.

We fix $\alpha > 0$ and define the function $W$ from $\mathbb{K}[X]$ by $$W : \begin{array}{ccc} \mathbb{K}[X] & \rightarrow & \mathbb{K}[X] \\ p & \mapsto & p(\alpha X) \end{array}$$
Show that $W$ is an automorphism of $\mathbb{K}[X]$.

We fix $\alpha > 0$ and define $W : p \mapsto p(\alpha X)$. We set $P = W \circ L \circ W^{-1}$ where $L$ is the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.
Show that $$P = \frac{1}{\alpha} D \circ \left(\frac{1}{\alpha} D - I\right)^{-1}$$

We fix $\alpha > 0$, define $W : p \mapsto p(\alpha X)$, and set $P = W \circ L \circ W^{-1}$ where $L$ is the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We have $P = \frac{1}{\alpha} D \circ \left(\frac{1}{\alpha} D - I\right)^{-1}$.
Show that $P$ is a delta endomorphism whose associated polynomial sequence $(p_n)_{n \in \mathbb{N}}$ satisfies $$\forall n \in \mathbb{N}, \quad p_n = \ell_n(\alpha X)$$

Let $L$ be the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$, and let $P = W \circ L \circ W^{-1}$ with $W : p \mapsto p(\alpha X)$.
Verify that $D = L \circ (L-I)^{-1}$ then that $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.

We denote by $T$ the unique automorphism satisfying, for all $n \in \mathbb{N}$, $T\ell_n = \frac{X^n}{n!}$ and we set $Q = T \circ P \circ T^{-1}$, where $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.
Show that $Q = D \circ (\alpha I + (1-\alpha)D)^{-1}$. Deduce that $Q$ is a delta endomorphism whose associated polynomial sequence $(r_n)_{n \in \mathbb{N}}$ satisfies $$\forall n \in \mathbb{N}^*, \quad r_n = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \frac{X^k}{k!}$$

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) \neq 0$. The purpose of this question is to show that there exists $r \in \mathbb { R } _ { + } ^ { * } , r \leqslant \rho$ such that $\frac { 1 } { f } \in \mathscr { D } _ { r } ( \mathbb { R } )$.
6a. Show that we can assume without loss of generality that $f ( 0 ) = 1$.
We now write $f ( t ) = \sum _ { i = 0 } ^ { \infty } a _ { i } t ^ { i }$ and we assume that $a _ { 0 } = 1$.
6b. Only in this sub-question, we assume that there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $g \in \mathscr { D } _ { r } ( \mathbb { R } )$ such that $f ( t ) g ( t ) = 1$ for all $t \in U _ { r }$. We write $g ( t ) = \sum _ { i = 0 } ^ { \infty } b _ { i } t ^ { i }$. Show that: $$\left\{ \begin{aligned} & b _ { 0 } = 1 \\ & \text { for } n \geqslant 1 , b _ { n } = - \left( b _ { 0 } a _ { n } + \ldots + b _ { n - 1 } a _ { 1 } \right) \end{aligned} \right.$$
We now define the sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ by the above recurrence formula.
6c. Show that there exists $c \in \mathbb { R } _ { + } ^ { * }$ such that $\left| a _ { n } \right| \leqslant c ^ { n }$ for all $n \in \mathbb { N }$.
6d. Show that $\left| b _ { n } \right| \leqslant ( 2 c ) ^ { n }$ for all $n \in \mathbb { N }$.
6e. Conclude.

Show that $\Delta$ is an endomorphism of $\mathbb{K}[X]$, where $$\Delta : \begin{cases} \mathbb{K}[X] \rightarrow \mathbb{K}[X] \\ P(X) \mapsto P(X+1) - P(X) \end{cases}$$

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 $d \in \mathbb{N}^{*}$.

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.

For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, the Bernoulli polynomial is defined by $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$. Show that, for all $n \in \mathbb{N}^{*}$, $B_{n}' = n B_{n-1}$.

Let $a \in \mathbf{Q}^{\times}$ be a nonzero rational number. Deduce from Theorem 1 that, for every nonzero polynomial $P \in \mathbf{Q}[x]$, we have $P(e^a) \neq 0$.
(Theorem 1: Let $r \geq 2$ be an integer. If $a_1, \ldots, a_r \in \mathbf{Q}$ are distinct rational numbers, then the real numbers $e^{a_1}, \ldots, e^{a_r}$ are linearly independent over $\mathbf{Q}$.)

Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients such that 0 is not a root. Show that there exists a unique power series $f \in \mathbf{Q}\llbracket x \rrbracket$ satisfying $Q \cdot f = 1$.
Show that if $Q$ has integer coefficients and its constant term $c_0$ equals 1 or $-1$, then this unique power series $f$ has integer coefficients.

Show that if 0 is not a pole of $P/Q \in \mathbf{Q}(x)$, then there exists a unique power series with rational coefficients $g \in \mathbf{Q}\llbracket x \rrbracket$ such that $P = Q \cdot g$.
Show that the map $P/Q \longmapsto g$ is compatible with addition and multiplication in $\mathbf{Q}(x)$ and in $\mathbf{Q}\llbracket x \rrbracket$, and that it sends the derivative $(P/Q)' = (P'Q - PQ')/Q^2$ to the derived power series $g'$.

Show that the power series $$\sum_{m=0}^{\infty} x^{m^2} = \sum_{n=0}^{\infty} c_n x^n$$ where $c_n = 1$ if $n$ is the square of an integer $m \geq 0$ and $c_n = 0$ otherwise, is not the power series expansion of a rational function.

Give a new proof, based on questions 12 and 13 above, of the fact that the power series $\sum_{m=0}^{\infty} x^{m^2}$ is not the expansion of a rational function.

Let $r \geq 2$ be an integer and $a_1, \ldots, a_r \in \mathbf{Q}$ be distinct rationals. Let $b_1, \ldots, b_r \in \mathbf{Q}^{\times}$ be nonzero rationals. Set $e^{a_i x} \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{a_i^n}{n!} x^n$ and consider the power series $$f(x) = \sum_{n=0}^{\infty} \frac{u_n}{n!} x^n \stackrel{\text{def}}{=} b_1 e^{a_1 x} + \cdots + b_r e^{a_r x}.$$ Show that the Laplace transform $\widehat{f}(x) = \sum_{n=0}^{\infty} u_n x^n$ is the power series expansion of the rational function $$\sum_{i=1}^{r} \frac{b_i}{1 - a_i x}.$$ Deduce that $f$ is not the zero power series.