Centrale-Supélec 2026 — Mathématiques 2 (PSI)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A — Un produit scalaire à poids (Q1 à Q8)

   α and β are two real numbers strictly greater than −1.

   • Q1, Q2 — convergence of Riemann integrals at one endpoint.
   • Q3 — convergence of ∫_{−1}^{1}(1−t)^α(1+t)^β f(t)dt for continuous f.
   • Q4 — this integral defines an inner product on R[X].
   • Q5, Q6 — the case α = β = −1/2.
   • Q7, Q8 — the case α = β = 0.
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2. Exercise 2 — Partie B — Autour des coefficients binomiaux (Q9, Q10)

   • Q9 — Vandermonde's identity, by a counting argument.
   • Q10 — comparison of the binomial coefficient C(n, k) and the generalised coefficient B_k^n.
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3. Exercise 3 — Partie C · I — Premières propriétés des polynômes P_k (Q11 à Q13)

   (P_0, …, P_n) is the orthonormal basis obtained by Gram-Schmidt from (1, X, …, X^n).

   • Q11 — P_k has degree k.
   • Q12 — P_k is orthogonal to all polynomials of degree < k.
   • Q13 — every polynomial of degree k orthogonal to Vect(1, …, X^{k−1}) is proportional to P_k.
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4. Exercise 4 — Partie C · II et III — Relation à trois termes et racines (Q14 à Q19)

   • Q14, Q15 — the existence of a recurrence relation linking P_{k+1}, P_k and P_{k−1}.
   • Q16 to Q19 — the roots of P_k in ]−1, 1[: their number m, the polynomial S built on them, and the conclusion.
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5. Exercise 5 — Partie D · I — Les polynômes de Jacobi (Q20 à Q24)

   g_k(x) = (1−x)^{α+k}(1+x)^{β+k} and J_k is defined from g_k^{(k)}.

   • Q20 — the expressions of g_0, g_1, then of J_0 and J_1.
   • Q21 — the p-th derivative of x ↦ (1−x)^j for real j.
   • Q22, Q23 — the expression of g_k^{(k)} by Leibniz's formula, then that of J_k.
   • Q24 — J_k is polynomial of degree k, and its leading coefficient.
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6. Exercise 6 — Partie D · II — Orthogonalité des polynômes de Jacobi (Q25 à Q28)

   • Q25 — convergence of two integrals, and a relation between them obtained by integration by parts.
   • Q26 — ⟨J_k, X^i⟩ = 0 for i ⩽ k−1.
   • Q27 — (J_0, …, J_n) is an orthogonal basis.
   • Q28 — how to deduce an orthonormal basis.
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7. Exercise 7 — Partie E · I — Les polynômes de Legendre (Q29 à Q33)

   Case α = β = 0: L_k = (1/(2^k k!))·[(X²−1)^k]^{(k)}.

   • Q29, Q30 — the first four polynomials and their roots.
   • Q31 — degree and leading coefficient, without using Part C.
   • Q32, Q33 — parity, then the values at 1 and −1.
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8. Exercise 8 — Partie E · II et III — Équation différentielle et majoration (Q34 à Q37)

   Here g_k(x) = (1−x²)^k.

   • Q34 — express (1−x²)g_k′(x) in terms of g_k.
   • Q35 — differentiate it k+1 times using Leibniz: L_k satisfies Legendre's equation.
   • Q36, Q37 — the auxiliary function f_k is monotonic, hence |L_k| ⩽ 1.
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9. Exercise 9 — Partie E · IV et V — Récurrence et fonction génératrice (Q38 à Q42)

   • Q38, Q39 — the three-term relation made explicit for the L_k.
   • Q40 — the radius of convergence of Σ L_k(x)t^k is at least 1.
   • Q41, Q42 — the differential equation satisfied by the sum S, then its closed form.
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