Consider the sequence $(v_n)_{n \geq 0}$ defined in terms of the coefficients $u_n$ by the formula $$v_n = n! \sum_{i=0}^{n} \frac{u_i}{i!}$$ and the power series $$v(x) = \sum_{n=0}^{\infty} v_n x^n \in \mathbf{Q}\llbracket x \rrbracket.$$ Show the equality of power series $$\sum_{n=0}^{\infty} (v_n - n v_{n-1}) x^n = \sum_{n=0}^{\infty} u_n x^n.$$

With the notation of question 18 and 19, show that the differential operator $L = -x^2 \left(\frac{d}{dx}\right) + (1-x)$ acts on $v(x)$ by $$(L \cdot v)(x) = \sum_{i=1}^{r} \frac{b_i}{1 - a_i x}.$$

With the notation of questions 18--20, deduce that if $v(x)$ is the power series expansion of a rational fraction $P/Q$, then every element of the non-empty set $\{1/a_i \mid a_i \neq 0\}$ is a pole of $P/Q$.

Let $\ell \geq 0$ be an integer. Show that there exists a polynomial $P_\ell \in \mathbf{Q}[x]$ of degree $< r(\ell+1)$ satisfying $$\sum_{n=0}^{\infty} n(n-1)\cdots(n-\ell+1)\, u_{n-\ell}\, x^n = \frac{P_\ell(x)}{\left(1 - s_1 x - \cdots - s_r x^r\right)^{\ell+1}}.$$

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.

Show that the set of quasi-polynomial functions forms a $\mathbb{C}$-vector space.

Show that if $P, Q : \mathbb{Z} \rightarrow \mathbb{C}$ are two quasi-polynomial functions such that $P(n) = Q(n)$ for all $n \geq 0$, then $P = Q$.

Show that a function $P : \mathbb{Z} \rightarrow \mathbb{C}$ is quasi-polynomial if and only if there exist an integer $m \in \mathbb{N}^*$ and $m$ polynomials $P_0, \ldots, P_{m-1}$ with complex coefficients such that for all $j \in \{0, \ldots, m-1\}$ and for all $n \in \mathbb{Z}$ congruent to $j$ modulo $m$, we have $P(n) = P_j(n)$.

Let $\omega$ be a root of unity and $p \in \mathbb{N}^*$. Let $\sum_{n=0}^{+\infty} R(n) x^n$ denote the power series expansion of $\frac{1}{(1 - \omega x)^p}$. Show that $R$ is a quasi-polynomial function then determine its degree and its leading coefficient.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\},$$ then we define the power series $F(x) = \sum_{n=0}^{\infty} P(n) x^n$.
Deduce that $P$ is a quasi-polynomial function.

Let $k \in \mathbb{N}^*$ and $(a_1, \ldots, a_k) \in (\mathbb{N}^*)^k$ a $k$-tuple of strictly positive integers. When $k \geq 2$, we assume they are coprime as a set. We define a function $P : \mathbb{N} \rightarrow \mathbb{C}$ by setting for all $n \in \mathbb{N}$: $$P(n) = \operatorname{Card}\left\{(n_1, \ldots, n_k) \in \mathbb{N}^k : n_1 a_1 + \cdots + n_k a_k = n\right\},$$ then we define the power series $F(x) = \sum_{n=0}^{\infty} P(n) x^n$.
Calculate the leading coefficient of $P$.