We define a sequence $(a_n)_{n \geqslant 1}$ by setting $$\forall n \in \mathbb{N}^*, \quad a_n = \frac{(-n)^{n-1}}{n!}.$$ We define, when possible, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$, with radius of convergence $R$. Let $W$ be the Lambert function defined in Part I (inverse of $f|_{[-1,+\infty[}$ where $f(x)=xe^x$). Using the results of Questions 31 and 32, deduce that $$\forall x \in ]-R, R[, \quad S(x) = W(x).$$

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Using the fact that $0 \leqslant \phi_x'(t) \leqslant \frac{x}{\mathrm{e}}$ for all $t \in \mathbb{R}$, deduce that $$\forall x \in [0, \mathrm{e}], \quad \forall n \in \mathbb{N}, \quad |w_n(x) - W(x)| \leqslant \left(\frac{x}{\mathrm{e}}\right)^n |1 - W(x)|.$$

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. For every real $a \in ]0, \mathrm{e}[$, justify that the sequence of functions $(w_n)$ converges uniformly on $[0, a]$ to the function $W$.

For every positive real $x$, we consider the function $\phi_x$ defined by $$\phi_x : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ t & \mapsto & x\exp(-x\exp(-t)) \end{array}$$ and a sequence of functions $(w_n)_{n \geqslant 0}$ on $\mathbb{R}^+$ defined by $$\forall x \in \mathbb{R}^+, \quad \begin{cases} w_0(x) = 1 \\ w_{n+1}(x) = \phi_x(w_n(x)) \end{cases}$$ Let $W$ be the Lambert function defined in Part I. Does the sequence of functions $(w_n)$ converge uniformly to $W$ on $[0, \mathrm{e}]$?

Deduce from this, for every natural integer $k$, an expression for $P ^ { ( k ) }$ as a function of $T$, $k$ and $P ^ { ( 0 ) }$.

Let $\left(u_n\right)_{n \in \mathbb{N}}$ be a hypergeometric sequence with associated polynomials $P$ and $Q$. Suppose that there exists a natural integer $n_0$ such that $P\left(n_0\right) = 0$ and, $\forall n \geqslant n_0, Q(n) \neq 0$. Justify that the sequence $\left(u_n\right)_{n \in \mathbb{N}}$ is zero from a certain rank onwards.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$. For every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that, for every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, $F(x) = 1 + x(F(x))^{2}$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\forall t \in I , g ( t ) ^ { 2 } = 2 g ( t ) - 4 t$.

We recall that the sequence $\left(\left(\sum_{k=1}^{n} k^{-1}\right) - \ln(n)\right)_{n \geqslant 1}$ converges. We denote $\gamma$ its limit. Let $n \in \mathbb{N}^*$. For $x \in ]0, +\infty[$, we set $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the sequence of functions $\left(\Gamma_n\right)_{n \geqslant 1}$ converges pointwise on $]0, +\infty[$ to a function $\Gamma$ from $]0, +\infty[$ to $]0, +\infty[$.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that the function $f : \left|]-\frac{1}{4}, \frac{1}{4}[ \begin{array}{cc} & \rightarrow \\ x & \mathbb{R} \\ & \mapsto 2xF(x) - 1 \end{array}\right.$ does not vanish.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$. For every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Show that the function $f : \left]-\frac{1}{4}, \frac{1}{4}\right[ \rightarrow \mathbb{R},\; x \mapsto 2xF(x) - 1$ does not vanish.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that there exists a function $\varepsilon : I \rightarrow \{ - 1,1 \}$ such that $$\forall t \in I , \quad g ( t ) = 1 + \varepsilon ( t ) \sqrt { 1 - 4 t } .$$

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that, for all $x \in ]0, +\infty[$, we have $\Gamma(x+1) = x\Gamma(x)$.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convention $C_{0} = 1$. For every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Determine, for every $x \in ]-\frac{1}{4}, \frac{1}{4}[$, an expression of $F(x)$ as a function of $x$.

For every integer $n \geqslant 1$, we denote $C_{n}$ the number of well-parenthesized words of length $2n$. We set by convenience $C_{0} = 1$. For every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, we set $F(x) = \sum_{k=0}^{+\infty} C_{k} x^{k}$.
Determine, for every $x \in \left]-\frac{1}{4}, \frac{1}{4}\right[$, an expression of $F(x)$ as a function of $x$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\varepsilon$ is continuous on $I \backslash \left\{ \frac { 1 } { 4 } \right\}$. Deduce $$\forall t \in I , \quad g ( t ) = 1 - \sqrt { 1 - 4 t } .$$

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the function $\Gamma$ is of class $\mathscr{C}^2$ and that, for all $x \in ]0, +\infty[$, $$(\ln(\Gamma))''(x) = \sum_{k=0}^{+\infty} \frac{1}{(x+k)^2}.$$

Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where $$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that $\lim_{x \rightarrow +\infty} (\ln(\Gamma))''(x) = 0$.

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Deduce that $\mathbb { P } ( T \neq 0 ) = 1 - \sqrt { 1 - 4 p ( 1 - p ) }$. Interpret this result when $p = \frac { 1 } { 2 }$.

Let $z \in D$. Show the convergence of the series $\sum _ { n \geq 1 } \frac { z ^ { n } } { n }$. Specify the value of its sum when $z \in ] - 1,1 [$. We denote
$$L ( z ) : = \sum _ { n = 1 } ^ { + \infty } \frac { z ^ { n } } { n }$$

Let $z \in D$. Show the convergence of the series $\sum _ { n \geq 1 } \frac { z ^ { n } } { n }$. Specify the value of its sum when $z \in ] - 1,1 [$. We denote
$$L ( z ) : = \sum _ { n = 1 } ^ { + \infty } \frac { z ^ { n } } { n }$$

Let $z \in D$. Show the convergence of the series $\sum_{n \geq 1} \frac{z^n}{n}$. Specify the value of its sum when $z \in ]-1,1[$. We denote $$L(z) := \sum_{n=1}^{+\infty} \frac{z^n}{n}$$

Let $f$ be a power series and $z$ a complex number such that $\hat{f}(|z|) < \infty$. Show then that the series $f$ converges at $z$ and that $|f(z)| \leqslant \hat{f}(|z|)$. Give an example where this inequality is strict.

Let $\ell$ be a strictly positive integer. We are given a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ of vectors in $\mathbb { R } ^ { \ell }$ such that the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$ converges.
(a) Show that the sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ is convergent.
(b) Let $v ^ { * }$ denote the limit of this sequence. Bound $\left\| v _ { n } - v ^ { * } \right\|$ by means of a remainder of the sum of the series $\sum _ { n } \left\| v _ { n + 1 } - v _ { n } \right\|$.

We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $g = f^\dagger$ the reciprocal series. Show that $\hat{g} \prec (1/\lambda)(I + \hat{F} \circ \hat{g})$, conclude using part C that $\rho(g) > 0$ if $\rho(f) > 0$.