Define $f _ { k } ( n )$ to be the sum of all possible products of $k$ distinct integers chosen from the set $\{ 1,2 , \ldots , n \}$, i.e., $$f _ { k } ( n ) = \sum _ { 1 \leq i _ { 1 } < i _ { 2 } < \ldots < i _ { k } \leq n } i _ { 1 } i _ { 2 } \ldots i _ { k }$$ a) For $k > 1$, write a recursive formula for the function $f _ { k }$, i.e., a formula for $f _ { k } ( n )$ in terms of $f _ { \ell } ( m )$, where $\ell < k$ or ($\ell = k$ and $m < n$). b) Show that $f _ { k } ( n )$, as a function of $n$, is a polynomial of degree $2k$. c) Express $f _ { 2 } ( n )$ as a polynomial in variable $n$.

Let $f , g , h$ be functions from $\mathbb { R }$ to $\mathbb { R }$ such that $$h ( f ( x ) + g ( y ) ) = x y$$ for all $x , y \in \mathbb { R }$. Show the following:
(A) $(2$ marks$)$ $h$ is surjective.
(B) $(3$ marks$)$ If $f$ is continuous then $f$ is strictly monotone.
(C) $(5$ marks$)$ There do not exist continuous functions $f , g , h$ satisfying $(*)$.

21. (12 points) Let $f _ { n } ( x )$ be the sum of the terms of the geometric sequence $1 , x , x ^ { 2 } , \cdots , x ^ { n }$, where $x > 0$, $n \in \mathrm {~N} , ~ n \geq 2$. (I) Prove that the function $\mathrm { F } _ { n } ( x ) = f _ { n } ( x ) - 2$ has exactly one zero in $\left( \frac { 1 } { 2 } , 1 \right)$ (denoted as $x _ { n }$), and $x _ { n } = \frac { 1 } { 2 } + \frac { 1 } { 2 } x _ { n } ^ { n + 1 }$; (II) Consider an arithmetic sequence with the same first term, last term, and number of terms as the above geometric sequence, with sum $g _ { n } ( x )$. Compare the sizes of $f _ { n } ( x )$ and $g _ { n } ( x )$, and provide a proof.

Choose one of questions 22, 23, or 24 to answer. If you do more than one, only the first one will be graded. Mark the box number of your chosen question with a 2B pencil on the answer sheet.

For a function $h \in C([-1,1])$, we denote by $\widetilde{h}$ the following $2\pi$-periodic function: $$\tilde{h} : \begin{cases} \mathbb{R} \rightarrow \mathbb{R} \\ \theta \mapsto h(\cos(\theta)) \end{cases}$$ For every integer $n \in \mathbb{N}$, $F_n(x) = \cos(n \arccos x)$ extended as a polynomial to $\mathbb{R}$.
Let $f \in C^\infty([-1,1])$.
Show that there exists a sequence $(\alpha_n(f))_{n \in \mathbb{N}}$ with rapid decay such that $$f(x) = \sum_{n=0}^{+\infty} \alpha_n(f) F_n(x)$$ for all $x \in [-1,1]$. Give an expression for $\alpha_n(f)$ in terms of $f$ and $n$.

Let $f$ be a complex function of class $C ^ { \infty }$ on $[ 0,1 ]$.
a) Show that for every integer $q \geqslant 1$ $$f ( 1 ) - f ( 0 ) = \sum _ { j = 1 } ^ { q } ( - 1 ) ^ { j + 1 } \left[ A _ { j } ( t ) f ^ { ( j ) } ( t ) \right] _ { 0 } ^ { 1 } + ( - 1 ) ^ { q } \int _ { 0 } ^ { 1 } A _ { q } ( t ) f ^ { ( q + 1 ) } ( t ) d t$$
b) Taking into account the relations found in the previous part, show that for every odd natural integer $q = 2 p + 1$ $$f ( 1 ) - f ( 0 ) = \frac { 1 } { 2 } \left( f ^ { \prime } ( 0 ) + f ^ { \prime } ( 1 ) \right) - \sum _ { j = 1 } ^ { p } a _ { 2 j } \left( f ^ { ( 2 j ) } ( 1 ) - f ^ { ( 2 j ) } ( 0 ) \right) - \int _ { 0 } ^ { 1 } A _ { 2 p + 1 } ( t ) f ^ { ( 2 p + 2 ) } ( t ) d t$$

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. Using question III.B.1, determine the Fourier coefficients of $\widetilde { A } _ { p }$: $$\widehat { A } _ { p } ( n ) = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \widetilde { A } _ { p } ( t ) e ^ { - i n x } d x$$

For every natural integer $p \geqslant 1$ and every real number $x$, we set $\widetilde { A } _ { p } ( x ) = A _ { p } \left( \frac { x } { 2 \pi } - \left[ \frac { x } { 2 \pi } \right] \right)$. For $p \in \mathbb { N } ^ { * }$, deduce that $a _ { 2 p } = A _ { 2 p } ( 0 ) = ( - 1 ) ^ { p + 1 } \frac { S ( 2 p ) } { 2 ^ { 2 p - 1 } \pi ^ { 2 p } }$.

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$. The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the orthonormal family defined in question II.E.
Express $s_n$ using the sequence of polynomials $\left(K_p\right)_{p \in \mathbb{N}}$.

In this question I.C.3, assume that $g$ is continuous, $2\pi$-periodic and of class $C^{1}$ piecewise. a) State without proof the theorem on Fourier series applicable to continuous, $2\pi$-periodic functions of class $C^{1}$ piecewise. b) Show that $f * g$ is $2\pi$-periodic and is the sum of its Fourier series. Specify the Fourier coefficients of $f * g$ using the Fourier coefficients of $g$ and integrals involving $f$.

Let $f, g \in L^{1}(\mathbb{R})$. Assume that $g$ is bounded. a) Show that $f * g$ is integrable on $\mathbb{R}$ and determine $\int_{\mathbb{R}} f * g$ in terms of $\int_{\mathbb{R}} f$ and $\int_{\mathbb{R}} g$. b) Show that $\widehat{f * g} = \hat{f} \times \hat{g}$.

Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set $$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$ For every real $t$ and every non-zero natural number $n$, show that $I_{n}(t) = \left(f * K_{n}\right)(t)$.

Let $f \in L^{1}(\mathbb{R})$ be such that $\hat{f} \in L^{1}(\mathbb{R})$. For every real $t$ and every non-zero natural number $n$, we set $$I_{n}(t) = \frac{1}{2\pi} \int_{\mathbb{R}} k_{n}(x) \hat{f}(-x) \mathrm{e}^{-\mathrm{i}tx} \mathrm{~d}x$$ Deduce, for every real $t$: $$f(t) = \frac{1}{2\pi} \int_{\mathbb{R}} \hat{f}(x) \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}x$$

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $\widetilde{a}_n = \frac{1}{n+1}\sum_{k=0}^n a_k$.
Construct using $\psi$ an example of a sequence $\left( a _ { n } \right) _ { n \geqslant 0 }$ satisfying hypothesis II.1 ($f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$) but not satisfying property II.3 ($\lim_{n\to\infty} \widetilde{a}_n = 1$).

By expressing $G _ { x } ( - t )$ in terms of $G _ { x } ( t )$, show that for $n$ in $\mathbb { Z } , \varphi _ { n } ( x ) \in \mathbb { R }$.

Express $G _ { x } ( t + \pi )$ and deduce the following equalities for $n$ in $\mathbb { Z }$ :
$$\varphi _ { n } ( - x ) = ( - 1 ) ^ { n } \varphi _ { n } ( x ) = \varphi _ { - n } ( x )$$
What can be said about the parity of $\varphi _ { n }$ for $n \in \mathbb { Z }$ ?

Using Euler's formula, justify that for $( n , k )$ in $\mathbb { N } \times \mathbb { N }$,
$$I _ { n , k } = \sum _ { m = 0 } ^ { k } \frac { A _ { m , k } } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { i t ( 2 m - k - n ) } \mathrm { d } t$$
with $A _ { m , k }$ constants to be determined.

Let $p$ be the function from $\mathbb { R }$ to $\mathbb { R }$ defined by $$p ( x ) = \sum _ { n = 1 } ^ { + \infty } \frac { \cos ( n x ) } { n \sqrt { n } }$$
Determine the Fourier series of $p$.

Let $\theta \in \mathbb { R } \backslash 2 \pi \mathbb { Z }$.
Show that, for all $n \in \mathbb { N } ^ { * }$, $$\sum _ { k = 1 } ^ { n } \frac { \mathrm { e } ^ { \mathrm { i } k \theta } } { k } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { \mathrm { i } \theta } \frac { 1 - \left( \mathrm { e } ^ { \mathrm { i } \theta } t \right) ^ { n } } { 1 - \mathrm { e } ^ { \mathrm { i } \theta } t } \mathrm { ~d} t$$

Let $r : \mathbb { R } \rightarrow \mathbb { R }$, be a $2 \pi$-periodic, odd function, such that $\forall \theta \in ] 0 , \pi ] , r ( \theta ) = \frac { \pi - \theta } { 2 }$.
Justify the existence and uniqueness of $r$.

Let $r : \mathbb { R } \rightarrow \mathbb { R }$, be a $2 \pi$-periodic, odd function, such that $\forall \theta \in ] 0 , \pi ] , r ( \theta ) = \frac { \pi - \theta } { 2 }$.
Determine the Fourier series of $r$.

Let $x \in ] - 1,1 [$.
Determine the Fourier series of the function $\widetilde { h } : \mathbb { R } \rightarrow \mathbb { R }$ defined by $\widetilde { h } ( \theta ) = \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right)$.
One may use the result from question II.A.2.

Show that $\forall x \in \mathbb { R }$ and $\forall n \in \mathbb { N } ^ { * }$ $$\prod _ { k = 1 } ^ { n - 1 } \left( x ^ { 2 } - 2 x \cos \frac { 2 k \pi } { n } + 1 \right) = \left( \sum _ { k = 0 } ^ { n - 1 } x ^ { k } \right) ^ { 2 }$$

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Show that for all $j \in \mathbf{N}$ and all $k \in \mathcal{T}_{j+1}$, there exists a unique integer $k' \in \mathcal{T}_{j}$ such that $$[k 2^{-j-1}, (k+1) 2^{-j-1}] \subset [k' 2^{-j}, (k'+1) 2^{-j}]$$ Specify the relationship between $k$ and $k'$.

The Chebyshev polynomials of the first kind satisfy $T_m \cdot T_n = \frac{1}{2}(T_{n+m} + T_{n-m})$ for $0 \leqslant m \leqslant n$, and $T_m \cdot U_{n-1} = \frac{1}{2}(U_{n+m-1} + U_{n-m-1})$ for $0 \leqslant m < n$.
For $m$ and $n$ natural integers such that $m \leqslant n$, we propose to determine the quotient $Q_{n,m}$ and the remainder $R_{n,m}$ of the Euclidean division of $T_n$ by $T_m$.
a) Suppose $m < n < 3m$. Show that $$Q_{n,m} = 2T_{n-m} \quad \text{and} \quad R_{n,m} = -T_{|n-2m|}$$
b) Determine $Q_{n,m}$ and $R_{n,m}$ when $n$ is of the form $(2p+1)m$ with $p \in \mathbb{N}^*$.
c) Suppose that $m > 0$ and that $n$ is not the product of $m$ by an odd integer. Show that there exists a unique integer $p \geqslant 1$ such that $|n - 2pm| < m$ and that $$Q_{n,m} = 2\left(T_{n-m} - T_{n-3m} + \cdots + (-1)^{p-1} T_{n-(2p-1)m}\right) \quad \text{and} \quad R_{n,m} = (-1)^p T_{|n-2pm|}$$

We consider a real $\lambda$ and the sequence $\left(u_k = \lambda^k\right)_{k \in \mathbb{N}}$. What is the sequence $\left(v_k\right)_{k \in \mathbb{N}}$ defined by formula $$v_k = \sum_{j=0}^{k} \binom{k}{j} u_j \quad \text{(I.1)}$$ Then verify formula $$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j \quad \text{(I.2)}$$