grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2024 x-ens-maths-d__mp

45 maths questions

Q1 Number Theory Irrationality and Transcendence Proofs View
For a strictly positive rational number $a \in \mathbf{Q}_{>0}$, let $\log(a)$ denote the unique real number satisfying $e^{\log a} = a$. Deduce from Theorem 1 that $\log(a)$ is irrational for every strictly positive rational number $a \neq 1$.
(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 $\mathbf{Q}$.)
Q2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
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 $\mathbf{Q}$.)
Q3 Sequences and Series Power Series Expansion and Radius of Convergence View
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.
Q4 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients such that 0 is not a root. Show that there exists a unique power series $f \in \mathbf{Q}\llbracket x \rrbracket$ satisfying $Q \cdot f = 1$.
Show that if $Q$ has integer coefficients and its constant term $c_0$ equals 1 or $-1$, then this unique power series $f$ has integer coefficients.
Q5 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show that if 0 is not a pole of $P/Q \in \mathbf{Q}(x)$, then there exists a unique power series with rational coefficients $g \in \mathbf{Q}\llbracket x \rrbracket$ such that $P = Q \cdot g$.
Show that the map $P/Q \longmapsto g$ is compatible with addition and multiplication in $\mathbf{Q}(x)$ and in $\mathbf{Q}\llbracket x \rrbracket$, and that it sends the derivative $(P/Q)' = (P'Q - PQ')/Q^2$ to the derived power series $g'$.
Q6 Number Theory Algebraic Number Theory and Minimal Polynomials View
Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients whose constant term equals 1. Show that there exists an integer $b \geq 1$ such that $Q(bx)$ has integer coefficients.
Q7 Sequences and series, recurrence and convergence Series convergence and power series analysis View
Show that if $f \in \mathbf{Q}\llbracket x \rrbracket$ is the power series expansion of a rational function with rational coefficients, then $f$ is globally bounded.
(A power series $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ is globally bounded if there exist integers $A, B \geq 1$ such that $A f(Bx) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} (B^n A c_n) x^n$ is a power series with integer coefficients.)
Q8 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show that the power series $$\sum_{m=0}^{\infty} x^{m^2} = \sum_{n=0}^{\infty} c_n x^n$$ where $c_n = 1$ if $n$ is the square of an integer $m \geq 0$ and $c_n = 0$ otherwise, is not the power series expansion of a rational function.
Q9 Taylor series Derive series via differentiation or integration of a known series View
Give an example of a power series expansion of a rational function whose antiderivative is not the expansion of a rational function.
(The antiderivative of a power series $f(x) = \sum_{n=0}^{\infty} c_n x^n$ is defined as $\int_0^x f(t)\,dt \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{c_n}{n+1} x^{n+1}$.)
Q10 Polynomial Division & Manipulation View
(More difficult question) Let $f$ be the power series expansion of a rational function $P/Q \in \mathbf{Q}(x)$ all of whose poles are rational numbers. Suppose that the antiderivative $\int_0^x f(t)\,dt$ is globally bounded. Show that $\int_0^x f(t)\,dt$ is then the power series expansion of a rational function in $\mathbf{Q}(x)$.
Q11 Polynomial Division & Manipulation View
Show the equality: for all $m \geq 0$ and $\mu \geq 0$, $$\left(\frac{d}{dx}\right)^{\mu} \cdot \left(x^m f\right) = \left(x^m \left(\frac{d}{dx}\right)^{\mu} + \sum_{i=1}^{\min(\mu,m)} \frac{m(m-1)\cdots(m-i+1)\,\mu(\mu-1)\cdots(\mu-i+1)}{i!} x^{m-i} \left(\frac{d}{dx}\right)^{\mu-i}\right) \cdot f.$$
Q12 Taylor series Recursive or implicit derivative computation for series coefficients View
Show that the power series expansion of every rational function with rational coefficients is a solution of a differential equation of order 1.
Q13 Taylor series Recursive or implicit derivative computation for series coefficients View
Show that a power series $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ is a solution of a differential equation if and only if there exist an integer $d \geq 0$ and polynomials not all zero $S_0, \ldots, S_d \in \mathbf{Z}[x]$ such that: for all $n \geq 0$, $$S_0(n) c_n + \cdots + S_d(n) c_{n+d} = 0.$$
Q14 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Give a new proof, based on questions 12 and 13 above, of the fact that the power series $\sum_{m=0}^{\infty} x^{m^2}$ is not the expansion of a rational function.
Q15 Taylor series Recursive or implicit derivative computation for series coefficients View
Show that the power series $$h(x) = \sum_{n=0}^{\infty} \frac{(2n)!(3n)!}{(n!)^5} x^n$$ is a solution of a differential equation, then make one explicit.
Q16 Number Theory Divisibility and Divisor Analysis View
Show that $h(x) = \sum_{n=0}^{\infty} \frac{(2n)!(3n)!}{(n!)^5} x^n$ has integer coefficients.
Q17 Differential equations Higher-Order and Special DEs (Proof/Theory) View
Show that a power series $f(x) = \sum_{n=0}^{\infty} \frac{c_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ is a solution of a differential equation if and only if its Laplace transform $$\widehat{f}(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} c_n x^n$$ is a solution of a differential equation.
Q18 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $r \geq 2$ be an integer and $a_1, \ldots, a_r \in \mathbf{Q}$ be distinct rationals. Let $b_1, \ldots, b_r \in \mathbf{Q}^{\times}$ be nonzero rationals. Set $e^{a_i x} \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{a_i^n}{n!} x^n$ and consider the power series $$f(x) = \sum_{n=0}^{\infty} \frac{u_n}{n!} x^n \stackrel{\text{def}}{=} b_1 e^{a_1 x} + \cdots + b_r e^{a_r x}.$$ Show that the Laplace transform $\widehat{f}(x) = \sum_{n=0}^{\infty} u_n x^n$ is the power series expansion of the rational function $$\sum_{i=1}^{r} \frac{b_i}{1 - a_i x}.$$ Deduce that $f$ is not the zero power series.
Q19 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
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 $$\sum_{n=0}^{\infty} (v_n - n v_{n-1}) x^n = \sum_{n=0}^{\infty} u_n x^n.$$
Q20 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
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 $$(L \cdot v)(x) = \sum_{i=1}^{r} \frac{b_i}{1 - a_i x}.$$
Q21 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
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 $P/Q$.
Q22 Taylor series Prove smoothness or power series expandability of a function View
Show that $v(x)$ is not the power series expansion of a rational fraction.
Q23 Number Theory Algebraic Number Theory and Minimal Polynomials View
Show that Theorem 1 is equivalent to the following statement:
Let $f(x) \in \mathbf{Q}\llbracket x \rrbracket$ be an exponential polynomial such that $f(1) = \sum_{i=1}^{s} P_i(1) e^{c_i}$ vanishes. Then $f(x)/(x-1)$ is still an exponential polynomial.
(An exponential polynomial is any power series with rational coefficients of the form $f(x) = \sum_{i=1}^{s} P_i(x) e^{c_i x}$, where $c_1, \ldots, c_s \in \mathbf{Q}$ are rationals and $P_1, \ldots, P_s \in \mathbf{Q}[x]$ are polynomials.)
Q24 Sequences and series, recurrence and convergence Series convergence and power series analysis View
Recall the hypothesis $b_1 e^{a_1} + \cdots + b_r e^{a_r} = 0$ of Proposition 1. Define rational numbers $s_1, \ldots, s_r$ by the formula $$(T - a_1) \cdots (T - a_r) = T^r - s_1 T^{r-1} - \cdots - s_{r-1} T - s_r.$$ Show the equality: for all $n \geq 0$, $$u_{n+r} = s_1 u_{n+r-1} + \cdots + s_r u_n.$$
Q25 Number Theory Divisibility and Divisor Analysis View
Let $D$ be a common denominator of the rational numbers $a_1, \ldots, a_r$ and let $A = \max(1, |a_1|, \ldots, |a_r|)$. Show that $D^n u_n \in \mathbf{Z}$ for all $n \geq 0$.
Q26 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View
Show that there exists a real number $c_1 > 0$ such that: for all $n \geq 0$, $$|v_n| \leq c_1 \frac{A^{n+1}}{n+1}.$$
Q27 Sequences and series, recurrence and convergence Series convergence and power series analysis View
For all integers $n, k \geq 0$, define the rational number $v_n(k)$ as the coefficient of degree $n$ in the power series $$\left(1 - s_1 x - \cdots - s_r x^r\right)^k v(x) = \sum_{n=0}^{\infty} v_n(k) x^n.$$ Show that $v(x)$ is the power series expansion of a rational fraction if and only if there exists an integer $k \geq 0$ such that $\sum_{n=0}^{\infty} v_n(k) x^n$ is a polynomial.
Q28 Sequences and series, recurrence and convergence Series convergence and power series analysis View
For all integers $n, k \geq 0$, define the rational number $v_n(k)$ as the coefficient of degree $n$ in the power series $$\left(1 - s_1 x - \cdots - s_r x^r\right)^k v(x) = \sum_{n=0}^{\infty} v_n(k) x^n.$$ Observe the equality: for all $n \geq r$ and $k \geq 0$, $$v_n(k+1) = v_n(k) - s_1 v_{n-1}(k) - \cdots - s_r v_{n-r}(k).$$
Q29 Number Theory Divisibility and Divisor Analysis View
Let $C = 1 + |s_1| + \cdots + |s_r|$. Show that $D^n v_n(k) \in \mathbf{Z}$ and that there exists a real number $c_2 > 0$ such that $$|v_n(k)| \leq c_2 A^n C^k \text{ for all } n \geq kr.$$
Q30 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
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 $$\sum_{n=0}^{\infty} n(n-1)\cdots(n-\ell+1)\, u_{n-\ell}\, x^n = \frac{P_\ell(x)}{\left(1 - s_1 x - \cdots - s_r x^r\right)^{\ell+1}}.$$
Q31 Sequences and Series Recurrence Relations and Sequence Properties View
Define two sequences $(w_{n,k})_{n,k \geq 0}$ and $(w_n(k))_{n,k \geq 0}$ by the formulas $$w_{n,k} = n! \sum_{i=0}^{n-k} \frac{u_i}{i!} \quad \text{and} \quad \sum_{n=0}^{\infty} w_n(k) x^n = \left(1 - s_1 x - \cdots - s_r x^r\right)^k \sum_{n=0}^{\infty} w_{n,k}\, x^n.$$ Show the equality $w_n(k) = v_n(k)$ for all $n$ and $k$ such that $n \geq kr$.
Q32 Number Theory Divisibility and Divisor Analysis View
Deduce that $k!$ divides $D^n v_n(k)$ for all $n$ and $k$ such that $n \geq kr$.
Q33 Sequences and Series Evaluation of a Finite or Infinite Sum View
Show the equality $$v_n(k) = \sum_{i=1}^{r} b_i e^{a_i} \int_{a_i}^{\infty} e^{-t} t^{n-kr} (t - a_1)^k \cdots (t - a_r)^k\, dt.$$
Q34 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View
Show that if $v_n(k)$ is non-zero and $n \geq kr$, then $$k! \leq |D^n v_n(k)| \leq c_2 (AD)^n C^k.$$
Q35 Sequences and Series Recurrence Relations and Sequence Properties View
Deduce that there exists an integer $k_0$ such that $$v_n(k) = 0 \text{ for all } k \geq k_0 \text{ and } kr \leq n \leq 10kr.$$
Q36 Sequences and Series Recurrence Relations and Sequence Properties View
Conclude that $v_n(k_0) = 0$ for all $n \geq k_0 r$.
Q37 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that if $f$ is a function $E$, then the numerical series $\sum_{n=0}^{\infty} \frac{b_n}{n!} \alpha^n$ converges for every real number $\alpha$. We denote its value by $f(\alpha)$.
Q38 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f$ be a function $E$ that is not a polynomial. Show that there exists $R > 0$ such that the numerical series $\sum_{n=0}^{\infty} b_n \alpha^n$ diverges for every real number $\alpha$ with $|\alpha| > R$.
Q39 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Recall that $\widehat{f}(x)$ denotes the Laplace transform $\widehat{f}(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} b_n x^n$. Which functions $E$ are such that $\widehat{f}$ is also a function $E$?
Q40 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Prove that functions $E$ are closed under addition and multiplication.
Q41 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f$ be an exponential polynomial (i.e., $f(x) = \sum_{i=1}^{s} P_i(x) e^{c_i x}$ with $c_i \in \mathbf{Q}$ and $P_i \in \mathbf{Q}[x]$). Show that $f$ is a function $E$ such that $\widehat{f}$ is the power series expansion of a rational fraction with rational poles.
Q42 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that if $\sum_{n=0}^{\infty} b_n x^n$ is the power series expansion of a rational fraction with rational poles, then $\sum_{n=0}^{\infty} \frac{b_n}{n!} x^n$ is a function $E$.
Q43 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Show that the Bessel function $$J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$$ is a function $E$ such that $\widehat{J}_0(x)$ satisfies the equation $(1 + x^2)\widehat{J}_0(x)^2 = 1$. Deduce that $J_0(x)$ is not an exponential polynomial.
Q44 Sequences and series, recurrence and convergence Series convergence and power series analysis View
Show that the real zeros of the Bessel function $J_0(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} \frac{(-1)^n}{(n!)^2} \left(\frac{x}{2}\right)^{2n}$ are simple, that is, if $J_0(\alpha) = 0$, then $J_0'(\alpha) \neq 0$.
Q45 Sequences and series, recurrence and convergence Series convergence and power series analysis View
A function $E$ (with rational coefficients) is a power series $f(x) = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ satisfying: (a) $f$ is a solution of a differential equation; (b) there exists a real number $C > 0$ such that $|b_n| \leq C^n$ and $\operatorname{denom}(b_0, \ldots, b_n) \leq C^n$ for all $n \geq 1$.
Let $f(x)$ be a function $E$ such that $f(1) = 0$. Show that the power series $f(x)/(x-1)$ is still a function $E$.