In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = - 2$ then, (II.1) admits infinitely many constant solutions and infinitely many unbounded solutions.

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Show that if $a = + 2$ then, (II.1) admits infinitely many 2-periodic solutions and infinitely many unbounded solutions.

In this subsection II.A, $a$ is a nonzero real number. We denote by Sol(II.1) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad z _ { k + 1 } + a z _ { k } + z _ { k - 1 } = 0$$ Suppose in this question that $p$ is an integer greater than or equal to 3. Give a value of $a \in ] - 2,2 [$ for which all solutions of equation (II.1) are $p$-periodic.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2). We set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the sequence $\left( W _ { k } \right) _ { k \in \mathbb { N } }$ is constant.

We assume that $p$ is an integer greater than or equal to 2, that $\left( a _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( b _ { k } \right) _ { k \in \mathbb { N } }$ are two sequences of real numbers that are $p$-periodic and that $\forall k \in \mathbb { N } , b _ { k } \neq 0$. We denote by Sol(II.2) the set of complex sequences $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ that satisfy the recurrence relation $$\forall k \in \mathbb { N } ^ { * } , \quad b _ { k } z _ { k + 1 } + a _ { k } z _ { k } + b _ { k - 1 } z _ { k - 1 } = 0$$ We fix $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$, two solution sequences of (II.2), and set for all $k \in \mathbb { N } , W _ { k } = b _ { k } \left( y _ { k } z _ { k + 1 } - z _ { k } y _ { k + 1 } \right)$. Show that the two sequences $\left( y _ { k } \right) _ { k \in \mathbb { N } }$ and $\left( z _ { k } \right) _ { k \in \mathbb { N } }$ form a basis of $\operatorname { Sol } ($ II.2 $)$ if and only if $W _ { 0 } \neq 0$.

Determine the limit of $\zeta(x)$ as $x$ tends to $+\infty$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Using the result of Q12, deduce an asymptotic equivalent of $f$ at $+\infty$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. For all $x \in \mathcal{D}_{f}$, verify that $x + k \in \mathcal{D}_{f}$, then calculate $f(x+k) - f(x)$.

Let $f$ be the function defined by $$f(x) = \sum_{n=1}^{+\infty} \left(\frac{1}{n+x} - \frac{1}{n}\right)$$ Let $k \in \mathbb{N}^{*}$. Using the result of Q14, deduce an asymptotic equivalent of $f$ at $-k$. What are the right and left limits of $f$ at $-k$?

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. The sequence $(a_n)$ is defined by $$\left\{ \begin{array} { l } a _ { 0 } = 1 \\ a _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { ( - 1 ) ^ { n - k } } { ( n - k ) ! } H _ { k } \left( \frac { n + k } { 2 } - 1 \right) \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Show that, for all $z \in \mathcal { D }$, the power series $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ converges.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { 0 } ( z ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } z ^ { n }$ and, subject to convergence, $$\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$$
Justify that, for all $p \in \mathbb { N } ^ { * }$ and all $\left. x \in \right] - 1,1 \left[ , \Phi _ { p } ( x ) = \varphi ^ { ( p ) } ( x ) \right.$ and that for all $p \in \mathbb { N } ^ { * }$ and all $z \in \mathcal { D }$, the power series $\sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$ converges.

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$ Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand as power series on $D(0,R)$. What can we deduce about the function $f$?

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ Throughout this part, $f$ denotes a function that expands in a power series on $D(0,R)$.
Show that $f$ is of class $\mathcal{C}^1$ on $D(0,R)$ and that its partial derivatives expand in power series on $D(0,R)$. What can we deduce about the function $f$?

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. We define the function $\varphi : \mathbb { R } \rightarrow \mathbb { R }$ by $$\begin{cases} \varphi ( x ) = \exp \left( \frac { - x } { \sqrt { 1 - x } } \right) & \text { if } x < 1 \\ \varphi ( x ) = 0 & \text { if } x \geqslant 1 \end{cases}$$
Justify that, for all $p \in \mathbb { N }$, the function $\varphi ^ { ( p ) }$ is bounded on $] - 1,1 [$.

A function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands as a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i}y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$, so that, for any $(x,y) \in D(0,R)$, $$u(x,y) \in \mathbb{R}, \quad v(x,y) \in \mathbb{R}, \quad f(x,y) = u(x,y) + \mathrm{i}v(x,y).$$ Show that $u$ and $v$ are harmonic functions on $D(0,R)$.

We say that a function $f$, defined on $D(0,R) \subset \mathbb{R}^2$ and with complex values, expands in a power series on $D(0,R)$ if there exists a complex sequence $(a_n)$ such that $$\forall (x,y) \in D(0,R), \quad f(x,y) = \sum_{n=0}^{+\infty} a_n (x + \mathrm{i} y)^n$$ We denote by $u$ and $v$ the real and imaginary parts of $f$, so that, for any $(x,y) \in D(0,R)$, $$u(x,y) \in \mathbb{R}, \quad v(x,y) \in \mathbb{R}, \quad f(x,y) = u(x,y) + \mathrm{i} v(x,y).$$ Show that $u$ and $v$ are harmonic functions on $D(0,R)$.

We denote $\mathcal { D }$ the open unit disk of $\mathbb { C } : \mathcal { D } = \{ z \in \mathbb { C } ; | z | < 1 \}$. For $z \in \mathcal { D }$, we denote $\Phi _ { p } ( z ) = \sum _ { n = 0 } ^ { + \infty } ( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } z ^ { n }$. We admit that the function $\Phi _ { p }$ is bounded on $\mathcal { D }$.
Let $r$ be a real number in the interval $] 0,1 [$. Demonstrate for all integers $n \geqslant 1$ and $p \geqslant 1$, that $$( n + p ) ( n + p - 1 ) \cdots ( n + 1 ) a _ { n + p } r ^ { n } = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \Phi _ { p } \left( r \mathrm { e } ^ { \mathrm { i } \theta } \right) \mathrm { e } ^ { - n \mathrm { i } \theta } \mathrm {~d} \theta$$

We consider a general balanced urn. For all real $x, u$ and $v$, we set $$H(x, u, v) = \sum_{n=0}^{+\infty} P_{n}(u,v) \frac{x^{n}}{n!}$$ defined on $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2}$ for $\rho$ sufficiently small.
Prove that $H$ admits a first-order partial derivative with respect to $u$ on the domain $D_{\rho}$, obtained by term-by-term differentiation with respect to $u$ of the expression for $H$.

In the general model of a Pólya urn, we consider the balanced urn model for which $b = c = 0$, so $a = d$. Each time we draw a ball, we add $a$ balls of its color to the urn. The initial composition is $a_{0}$ white balls and $b_{0}$ black balls. The function $G$ is defined on $U = \{(x,u,v) \in \mathbb{R} \times \mathbb{R}_{+}^{*} \times \mathbb{R}_{+}^{*} ; axu^{a} < 1, axv^{a} < 1\}$ by $$G(x,u,v) = u^{a_{0}} v^{b_{0}} (1 - axu^{a})^{-a_{0}/a} (1 - axv^{a})^{-b_{0}/a}$$ We use the notation $D_{\rho} = ]-\rho, \rho[ \times ]0,2[^{2}$.
Using the preliminary results, prove that there exists $\rho > 0$ such that $D_{\rho} \subset U$ and, for all $(x,u,v) \in D_{\rho}$, $$G(x,u,v) = \sum_{n=0}^{+\infty} Q_{n}(u,v) \frac{x^{n}}{n!}$$ where $Q_{n}$ is a polynomial function of two variables to be specified.

Conclude that $$\frac { E \left( N _ { n } \right) } { n } \underset { n \rightarrow + \infty } { \longrightarrow } P ( R = + \infty ) .$$ One may admit and use Cesàro's theorem: if $\left( u _ { n } \right) _ { n \in \mathbb{N}^{*} }$ is a real sequence converging to the real number $\ell$, then $$\frac { 1 } { n } \sum _ { k = 1 } ^ { n } u _ { k } \underset { n \rightarrow + \infty } { \longrightarrow } \ell .$$

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, $S(x) = \sum_{n=1}^{+\infty} a_n x^n$. Determine the radius of convergence $R$ of the power series $\sum_{n \geqslant 1} a_n x^n$.

Let $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ be a sequence of reals such that the series $\sum (a_n)^2$ is convergent. Show that the radius of convergence of the power series $\sum a_n t^n$ is greater than or equal to 1.

Let $(a_n)_n \in \mathbb{R}^{\mathbb{N}}$ be a sequence of reals such that the series $\sum (a_n)^2$ is convergent. Show that the radius of convergence of the power series $\sum a_n t^n$ is greater than or equal to 1.

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 $R$. Prove that the function $S$ is defined and continuous on $[-R, R]$.

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 $R$. Prove that $$\forall x \in ]-R, R[, \quad x(1 + S(x))S'(x) = S(x).$$ One may use the result from Question 26.