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 $$n q_n(X) = X (Q')^{-1}\left(q_{n-1}\right)$$

We apply the results of question 40 to the endomorphism $L$ defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We denote by $(\ell_n)_{n \in \mathbb{N}}$ its associated sequence of polynomials.
Verify that, for $n \in \mathbb{N}^*$, $$\ell_n' = \ell_{n-1}' - \ell_{n-1}$$ and $$X\ell_n'' - X\ell_n' + n\ell_n = 0$$ and $$\ell_n(X) = \sum_{k=1}^n (-1)^k \binom{n-1}{k-1} \frac{X^k}{k!}$$

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 $$\forall n \in \mathbb{N}^*, \quad r_n = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \frac{X^k}{k!}$$

Using the results of the previous questions, conclude that $$\forall n \in \mathbb{N}^*, \quad \ell_n(\alpha X) = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \ell_k(X)$$

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

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

Show that for all $\theta \in ] - \pi ; \pi [$, the function $f$ defined by
$$\begin{aligned} f : ] 0 ; + \infty [ & \longrightarrow \mathbf { C } \\ t & \longmapsto \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \end{aligned}$$
is defined and integrable on $] 0 ; + \infty [$, where $x$ is a fixed element of $]0;1[$.

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

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 $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ be a power series with radius of convergence $R \geqslant 1$ and sum $f$. We denote $$\Delta _ { \theta _ { 0 } } = \left\{ z \in \mathbb { C } ; | z | < 1 \text { and } \exists \rho > 0 , \exists \theta \in \left[ - \theta _ { 0 } , \theta _ { 0 } \right] , z = 1 - \rho e ^ { i \theta } \right\}$$ for $\theta _ { 0 } \in [ 0 , \pi / 2 [$.
The purpose of this question is to prove that $$\left( \sum _ { n \geqslant 0 } a _ { n } \text { converges } \right) \Rightarrow \left( \lim _ { \substack { z \rightarrow 1 \\ z \in \Delta _ { \theta _ { 0 } } } } f ( z ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \right) \qquad \text{(Abel)}$$
(a) Prove (Abel) for $R > 1$.
From now on, we assume that $R = 1$ and that $\sum _ { n \geqslant 0 } a _ { n }$ converges, and we are given a $\theta _ { 0 } \in [ 0 , \pi / 2 [$.
(b) Prove that for all $N \in \mathbb { N } ^ { * }$ and $z \in \mathbb { C } , | z | < 1$, we have $$\sum _ { n = 0 } ^ { N } a _ { n } z ^ { n } - S _ { N } = ( z - 1 ) \sum _ { n = 0 } ^ { N - 1 } R _ { n } z ^ { n } - R _ { N } \left( z ^ { N } - 1 \right)$$
(c) Deduce that for all $z \in \mathbb { C } , | z | < 1$, we have $$f ( z ) - S = ( z - 1 ) \sum _ { n = 0 } ^ { + \infty } R _ { n } z ^ { n }$$
(d) Let $\varepsilon > 0$. Prove that there exists $N _ { 0 } \in \mathbb { N }$ such that for all $z \in \mathbb { C } , | z | < 1$ $$| f ( z ) - S | \leqslant | z - 1 | \sum _ { n = 0 } ^ { N _ { 0 } } \left| R _ { n } \right| + \varepsilon \frac { | z - 1 | } { 1 - | z | }$$
(e) Prove that there exists $\rho \left( \theta _ { 0 } \right) > 0$ such that for all $z \in \Delta _ { \theta _ { 0 } }$ of the form $z = 1 - \rho e ^ { i \theta }$ with $0 < \rho \leqslant \rho \left( \theta _ { 0 } \right)$, we have $$\frac { | z - 1 | } { 1 - | z | } \leqslant \frac { 2 } { \cos \left( \theta _ { 0 } \right) }$$ Deduce (Abel).

Prove that $$\sum _ { n = 0 } ^ { + \infty } \frac { ( - 1 ) ^ { n } } { 2 n + 1 } = \frac { \pi } { 4 }$$

Exhibit a power series $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ with radius of convergence 1 and sum $f$, such that $f ( z )$ converges when $z \rightarrow 1 , | z | < 1$ and such that the series $\sum _ { n \geqslant 0 } a _ { n }$ does not converge.

Let $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ be a power series with radius of convergence 1 and sum $f$. Let $S \in \mathbb { C }$. The purpose of this question is to prove that $$\left( \lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S \text { and } a _ { n } = o \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \sum _ { n \geqslant 0 } a _ { n } \text { converges and } \sum _ { n = 0 } ^ { + \infty } a _ { n } = S \right) . \quad \text{(Weak Tauberian)}$$
In the rest of this question we suppose that $\lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S$ and that $a _ { n } = o \left( \frac { 1 } { n } \right)$.
(a) Prove that for all $n \in \mathbb { N } ^ { * }$ and $x \in \left] 0,1 \right[$, we have $$\left| S _ { n } - f ( x ) \right| \leqslant ( 1 - x ) \sum _ { k = 1 } ^ { n } k \left| a _ { k } \right| + \frac { \sup _ { k > n } \left( k \left| a _ { k } \right| \right) } { n ( 1 - x ) }$$
(b) Deduce (Weak Tauberian) by specifying $x = x _ { n } = 1 - 1 / n$ for $n \in \mathbb { N } ^ { * }$.

Let $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ be a power series with radius of convergence 1 and sum $f$. Let $S \in \mathbb { C }$. The purpose of this question is to prove that $$\left( \lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S \text { and } a _ { n } = O \left( \frac { 1 } { n } \right) \right) \Rightarrow \left( \sum _ { n \geqslant 0 } a _ { n } \text { converges and } \sum _ { n = 0 } ^ { + \infty } a _ { n } = S \right) . \quad \text{(Strong Tauberian)}$$
(a) Prove that, without loss of generality, we can assume that $S = 0$.
We now suppose that $\lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } f ( x ) = S$ and that $a _ { n } = O \left( \frac { 1 } { n } \right)$, with $S = 0$.
(b) We define $\Theta$ as follows $$\Theta = \left\{ \theta : [ 0,1 ] \rightarrow \mathbb { R } ; \forall x \in \left[ 0,1 \left[ , \sum _ { n \geqslant 0 } a _ { n } \theta \left( x ^ { n } \right) \text { converges and } \lim _ { x \rightarrow 1 ^ { - } } \sum _ { n = 0 } ^ { + \infty } a _ { n } \theta \left( x ^ { n } \right) = 0 \right\} . \right. \right.$$ Prove that $\Theta$ is a vector space over $\mathbb { R }$.
(c) Let $P \in \mathbb { R } [ X ]$ such that $P ( 0 ) = 0$. Prove that $P \in \Theta$.
(d) Prove that $$\forall P \in \mathbb { R } [ X ] , \quad \lim _ { \substack { x \rightarrow 1 ^ { - } \\ x \in \mathbb { R } } } ( 1 - x ) \cdot \sum _ { n = 0 } ^ { + \infty } x ^ { n } P \left( x ^ { n } \right) = \int _ { 0 } ^ { 1 } P ( t ) d t$$
We define the function $g : \mathbb { R } \rightarrow \mathbb { R }$ by $$g ( x ) = \begin{cases} 1 & \text { if } x \in [ 1 / 2,1 ] \\ 0 & \text { otherwise } \end{cases}$$
(e) Prove that to establish (Strong Tauberian), it suffices to prove that $g \in \Theta$.
(f) Let $$h ( x ) = \begin{cases} - 1 & \text { if } x = 0 \\ \frac { g ( x ) - x } { x ( 1 - x ) } & \text { if } x \in ] 0,1 [ \\ 1 & \text { if } x = 1 \end{cases}$$ Given $\varepsilon > 0$, prove that there exist $s _ { 1 } , s _ { 2 } \in \mathcal { C } ^ { 0 } ( [ 0,1 ] )$ satisfying $$s _ { 1 } \leqslant h \leqslant s _ { 2 } \text { and } \int _ { 0 } ^ { 1 } \left( s _ { 2 } ( x ) - s _ { 1 } ( x ) \right) d x \leqslant \varepsilon$$ Represent graphically $h$ and two such functions $s _ { 1 } , s _ { 2 }$.
From now on, $\varepsilon > 0 , s _ { 1 }$ and $s _ { 2 }$ are fixed.
(g) Prove that there exist $T _ { 1 } , T _ { 2 } \in \mathbb { R } [ X ]$ such that $$\sup _ { x \in [ 0,1 ] } \left| T _ { 1 } ( x ) - s _ { 1 } ( x ) \right| \leqslant \varepsilon \quad \text { and } \quad \sup _ { x \in [ 0,1 ] } \left| T _ { 2 } ( x ) - s _ { 2 } ( x ) \right| \leqslant \varepsilon$$
We set, for all $x \in [ 0,1 ]$, $$P _ { 1 } ( x ) = x + x ( 1 - x ) \left( T _ { 1 } ( x ) - \varepsilon \right) , \quad P _ { 2 } ( x ) = x + x ( 1 - x ) \left( T _ { 2 } ( x ) + \varepsilon \right) \quad \text{and} \quad Q ( x ) = \frac { P _ { 2 } ( x ) - P _ { 1 } ( x ) } { x ( 1 - x ) }$$
(h) Prove that $$P _ { 1 } ( 0 ) = P _ { 2 } ( 0 ) = 0 , \quad P _ { 1 } ( 1 ) = P _ { 2 } ( 1 ) = 1 , \quad P _ { 1 } \leqslant g \leqslant P _ { 2 } \quad \text{and} \quad 0 \leqslant \int _ { 0 } ^ { 1 } Q ( x ) d x \leqslant 5 \varepsilon$$
(i) Prove that there exists $M > 0$ such that for all $x \in ] 0,1 [$, $$\left| \sum _ { n = 0 } ^ { + \infty } a _ { n } g \left( x ^ { n } \right) - \sum _ { n = 0 } ^ { + \infty } a _ { n } P _ { 1 } \left( x ^ { n } \right) \right| \leqslant M ( 1 - x ) \sum _ { n = 1 } ^ { + \infty } x ^ { n } Q \left( x ^ { n } \right)$$
(j) Conclude.

Let $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ be a power series whose coefficients $c_n$ are integers. Show that if there exists a real number $\alpha \geq 1$ such that the numerical series $\sum_{n=0}^{\infty} c_n \alpha^n$ converges, then $f$ is a polynomial.

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

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 $A^k = 0_n$).