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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
2023 centrale-maths1__official

44 maths questions

Q1 Groups Automorphism and Endomorphism Structure View
Let $a \in \mathbb{K}$. For all $p \in \mathbb{K}[X]$, we set $E_a(p) = E_a p = p(X+a)$.
Show that $E_a$ is an automorphism of $\mathbb{K}[X]$.
Q3 Polynomial Division & Manipulation View
To every $p \in \mathbb{R}[X]$, we associate the function $J(p) = Jp$ from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad J(p)(x) = Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$$
Show that $J$ preserves degree and that $J$ is invertible.
Q4 Reduction Formulae Compute a Base Case or Specific Value of a Parametric Integral View
To every $p \in \mathbb{K}[X]$, we associate the function $L(p) = Lp$ from $\mathbb{K}$ to $\mathbb{K}$ defined by $$\forall x \in \mathbb{K}, \quad L(p)(x) = Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$$
Show that $\int_0^{+\infty} \mathrm{e}^{-t} t^k\,\mathrm{d}t$ exists for all $k \in \mathbb{N}$ and calculate its value.
Q7 Groups Algebra and Subalgebra Proofs View
Show that the set of shift-invariant endomorphisms of $\mathbb{K}[X]$ is a subalgebra of $\mathcal{L}(\mathbb{K}[X])$. Is the set of delta endomorphisms of $\mathbb{K}[X]$ closed under addition? under composition?
Q8 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $\left(a_k\right)_{k \in \mathbb{N}}$ be a sequence of elements of $\mathbb{K}$. For every polynomial $p \in \mathbb{K}[X]$, show that the expression $$\sum_{k=0}^{+\infty} a_k D^k p$$ makes sense and defines a polynomial of $\mathbb{K}[X]$.
Q9 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Show that, for every sequence $\left(a_k\right)_{k \in \mathbb{N}}$ of elements of $\mathbb{K}$, $\sum_{k=0}^{+\infty} a_k D^k$ is a shift-invariant endomorphism.
Q10 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $\left(a_k\right)_{k \in \mathbb{N}}$ and $\left(b_k\right)_{k \in \mathbb{N}}$ be sequences of elements of $\mathbb{K}$ such that $\sum_{k=0}^{+\infty} a_k D^k = \sum_{k=0}^{+\infty} b_k D^k$.
Show that, for all $k \in \mathbb{N}$, $a_k = b_k$.
Q11 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
For every $n \in \mathbb{N}$, define the polynomial $q_n = \frac{X^n}{n!}$. Let $T$ be an endomorphism of $\mathbb{K}[X]$.
Show that $T$ is a shift-invariant endomorphism if, and only if, $$T = \sum_{k=0}^{+\infty} \left(T q_k\right)(0) D^k$$
Q12 Matrices Matrix Algebra and Product Properties View
Show that two shift-invariant endomorphisms of $\mathbb{K}[X]$ commute.
Q13 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View
For all $p \in \mathbb{K}[X]$ non-zero and $a \in \mathbb{K}$, show, using question 11, that $$p(X+a) = \sum_{k=0}^{\deg(p)} \frac{a^k}{k!} p^{(k)}$$ where $p^{(k)}$ denotes the $k$-th derivative of the polynomial $p$. Recognize this formula.
Q14 Integration by Parts Integral Involving a Parameter or Operator Identity View
For $p \in \mathbb{K}[X]$, express $Jp$ in terms of the derivatives $p^{(k)}$ ($k \in \mathbb{N}$) of $p$, where $J$ is defined by $Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$.
Q15 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View
Prove that the endomorphism $D - I$ is invertible and express $L$ in terms of $(D-I)^{-1}$, where $L$ is defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.
Q16 Proof Proof That a Map Has a Specific Property View
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$. We recall that the degree of the zero polynomial is by convention equal to $-1$.
Show that there exists a natural number $n(T)$ such that, for every polynomial $p \in \mathbb{K}[X]$, $$\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$$
Q17 Proof Deduction or Consequence from Prior Results View
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$, and let $n(T)$ be the natural number such that $\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$ for every $p \in \mathbb{K}[X]$.
Deduce $\ker(T)$ in terms of $n(T)$.
Q18 Proof Proof of Equivalence or Logical Relationship Between Conditions View
Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$.
Show that the following three assertions are equivalent:
  1. [(1)] $T$ is invertible;
  2. [(2)] $T1 \neq 0$;
  3. [(3)] $\forall p \in \mathbb{K}[X], \deg(Tp) = \deg(p)$.
Q21 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
Show that there exists a unique shift-invariant and invertible endomorphism $U$ such that $T = D \circ U$. Specify $U$ in the case $T = D$, then in the case $T = L$.
Q22 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
For every polynomial $p \in \mathbb{K}[X]$ non-zero, verify that $\deg(Tp) = \deg(p) - 1$. Deduce $\ker(T)$ and the spectrum of $T$.
Q23 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, we denote by $T_n$ the restriction of $T$ to $\mathbb{K}_n[X]$.
Show that $T_n$ is an endomorphism of $\mathbb{K}_n[X]$. Is it diagonalizable?
Q24 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, let $T_n$ denote the restriction of $T$ to $\mathbb{K}_n[X]$.
Determine $\operatorname{Im}(T_n)$ in terms of $n \in \mathbb{N}$ and deduce that $T$ is surjective.
Q25 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We wish to show that, for every delta endomorphism $Q$, there exists a unique sequence of polynomials $(q_n)_{n \in \mathbb{N}}$ of $\mathbb{K}[X]$ such that:
  • $q_0 = 1$;
  • $\forall n \in \mathbb{N}, \deg(q_n) = n$;
  • $\forall n \in \mathbb{N}^*, q_n(0) = 0$;
  • $\forall n \in \mathbb{N}^*, Q q_n = q_{n-1}$.

Let $Q$ be a delta endomorphism. Show the existence and uniqueness of the sequence $(q_n)_{n \in \mathbb{N}}$ of polynomials associated with $Q$.
Q26 Sequences and Series Functional Equations and Identities via Series View
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Show that, for every natural number $n$, $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$
Q27 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $(q_n)_{n \in \mathbb{N}}$ be a sequence of polynomials of $\mathbb{K}[X]$ such that $\forall n \in \mathbb{N}, \deg(q_n) = n$ and $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$
Show that there exists a unique delta endomorphism $Q$ for which $(q_n)_{n \in \mathbb{N}}$ is the associated sequence of polynomials.
Q28 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number.
Show that the family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.
Q29 Matrices Determinant and Rank Computation View
Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number. The family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.
According to question 23, $Q$ induces an endomorphism of $\mathbb{K}_n[X]$ denoted $Q_n$. Give its matrix in the previous basis. Deduce its trace, its determinant and its characteristic polynomial.
Q30 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
For $Q = D$, verify that $$\forall n \in \mathbb{N}, \quad q_n = \frac{X^n}{n!}$$
Q31 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
For $Q = E_1 - I$, verify that $$\forall n \in \mathbb{N}^*, \quad q_n = \frac{X(X-1)\cdots(X-n+1)}{n!}$$
Q32 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Prove that, for all $p \in \mathbb{K}[X]$, the expression $\sum_{k=0}^{+\infty} (Q^k p)(0) q_k$ makes sense and defines a polynomial of $\mathbb{K}[X]$, then that $$p = \sum_{k=0}^{+\infty} (Q^k p)(0) q_k$$
Q33 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Deduce that, for every shift-invariant endomorphism $T$, we have $$T = \sum_{k=0}^{+\infty} (T q_k)(0) Q^k$$
Q34 Sequences and Series Functional Equations and Identities via Series View
By choosing $Q = E_1 - I$, prove that, if $p$ is a non-constant polynomial, then $$p'(X) = \sum_{k=1}^{\deg(p)} \frac{1}{k} \left( \sum_{j=0}^k (-1)^{j+1} \binom{k}{j} p(X+j) \right)$$ This is the formula for numerical differentiation of polynomials.
Q35 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
Show that, if there exists $(a_n)_{n \in \mathbb{N}}$ a sequence of scalars such that $T = \sum_{k=0}^{+\infty} a_k D^k$, then $T' = \sum_{k=1}^{+\infty} k a_k D^{k-1}$.
Q36 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
If $T$ is a shift-invariant endomorphism, show that $T'$ is still a shift-invariant endomorphism.
Q37 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
If $T$ is an endomorphism of $\mathbb{K}[X]$, its Pincherle derivative $T'$ is defined by $$\forall p \in \mathbb{K}[X], \quad T'(p) = T(Xp) - XT(p)$$
If $T$ is a delta endomorphism, show that $T'$ is a shift-invariant and invertible endomorphism.
Q38 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $S$ and $T$ be two endomorphisms of $\mathbb{K}[X]$. The Pincherle derivative of an endomorphism $T$ is defined by $T'(p) = T(Xp) - XT(p)$.
Verify that $(S \circ T)' = S' \circ T + S \circ T'$.
Q39 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism. There exists a unique shift-invariant and invertible endomorphism $U$ such that $Q = D \circ U$. We denote by $(q_n)_{n \in \mathbb{N}}$ the sequence of polynomials associated with $Q$.
Prove that, for all $n \in \mathbb{N}^*$, we have $$\left(Q' \circ U^{-n-1}\right)\left(X^n\right) = X U^{-n}\left(X^{n-1}\right)$$
Q40 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism with $Q = D \circ U$ where $U$ is the unique shift-invariant and invertible endomorphism. We denote by $(q_n)_{n \in \mathbb{N}}$ the sequence of polynomials associated with $Q$.
Deduce that, for all $n \in \mathbb{N}^*$, $$n! q_n(X) = X U^{-n}\left(X^{n-1}\right)$$ then that $$n q_n(X) = X (Q')^{-1}\left(q_{n-1}\right)$$
Q41 Sequences and Series Recurrence Relations and Sequence Properties View
We apply the results of question 40 to the endomorphism $L$ defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We denote by $(\ell_n)_{n \in \mathbb{N}}$ its associated sequence of polynomials.
Verify that, for $n \in \mathbb{N}^*$, $$\ell_n' = \ell_{n-1}' - \ell_{n-1}$$ and $$X\ell_n'' - X\ell_n' + n\ell_n = 0$$ and $$\ell_n(X) = \sum_{k=1}^n (-1)^k \binom{n-1}{k-1} \frac{X^k}{k!}$$
Q42 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Show that there exists a unique invertible endomorphism $T$ such that $$\forall n \in \mathbb{N}, \quad T q_n = \frac{X^n}{n!}$$
Q43 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$, and let $T$ be the unique invertible endomorphism such that $T q_n = \frac{X^n}{n!}$ for all $n \in \mathbb{N}$.
Also show that $D = T \circ Q \circ T^{-1}$.
Q44 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We fix $\alpha > 0$ and define the function $W$ from $\mathbb{K}[X]$ by $$W : \begin{array}{ccc} \mathbb{K}[X] & \rightarrow & \mathbb{K}[X] \\ p & \mapsto & p(\alpha X) \end{array}$$
Show that $W$ is an automorphism of $\mathbb{K}[X]$.
Q45 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We fix $\alpha > 0$ and define $W : p \mapsto p(\alpha X)$. We set $P = W \circ L \circ W^{-1}$ where $L$ is the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.
Show that $$P = \frac{1}{\alpha} D \circ \left(\frac{1}{\alpha} D - I\right)^{-1}$$
Q46 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We fix $\alpha > 0$, define $W : p \mapsto p(\alpha X)$, and set $P = W \circ L \circ W^{-1}$ where $L$ is the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$. We have $P = \frac{1}{\alpha} D \circ \left(\frac{1}{\alpha} D - I\right)^{-1}$.
Show that $P$ is a delta endomorphism whose associated polynomial sequence $(p_n)_{n \in \mathbb{N}}$ satisfies $$\forall n \in \mathbb{N}, \quad p_n = \ell_n(\alpha X)$$
Q47 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
Let $L$ be the endomorphism defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$, and let $P = W \circ L \circ W^{-1}$ with $W : p \mapsto p(\alpha X)$.
Verify that $D = L \circ (L-I)^{-1}$ then that $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.
Q48 Sequences and Series Properties and Manipulation of Power Series or Formal Series View
We denote by $T$ the unique automorphism satisfying, for all $n \in \mathbb{N}$, $T\ell_n = \frac{X^n}{n!}$ and we set $Q = T \circ P \circ T^{-1}$, where $P = L \circ (\alpha I + (1-\alpha)L)^{-1}$.
Show that $Q = D \circ (\alpha I + (1-\alpha)D)^{-1}$. Deduce that $Q$ is a delta endomorphism whose associated polynomial sequence $(r_n)_{n \in \mathbb{N}}$ satisfies $$\forall n \in \mathbb{N}^*, \quad r_n = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \frac{X^k}{k!}$$
Q49 Sequences and Series Functional Equations and Identities via Series View
Using the results of the previous questions, conclude that $$\forall n \in \mathbb{N}^*, \quad \ell_n(\alpha X) = \sum_{k=1}^n \binom{n-1}{k-1} \alpha^k (1-\alpha)^{n-k} \ell_k(X)$$