Theorems · Theorem · commutative algebra

Polynomial.Chebyshev.one_sub_X_sq_mul_iterate_derivative_T_eq_poly_in_T

∀ {R : Type u_1} [inst : CommRing R] (n : ℤ) (k : ℕ),
  (1 - Polynomial.X ^ 2) * (⇑Polynomial.derivative)^[k + 2] (Polynomial.Chebyshev.T R n) =
    (2 * ↑k + 1) * Polynomial.X * (⇑Polynomial.derivative)^[k + 1] (Polynomial.Chebyshev.T R n) -
      (↑n ^ 2 - ↑k ^ 2) * (⇑Polynomial.derivative)^[k] (Polynomial.Chebyshev.T R n)

Defined in: Mathlib.RingTheory.Polynomial.Chebyshev
Cited by: 1 results in Mathlib
Foundations: Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
RingHom.idstatement · cited by 18,349
CommRingstatement and proof · cited by 17,173
LinearMapstatement · cited by 10,215
Polynomialstatement and proof · cited by 5,681
mul_oneproof · cited by 3,885
one_mulproof · cited by 2,841
add_zeroproof · cited by 2,707
Nat.cast_oneproof · cited by 2,501
mul_commproof · cited by 2,262
Polynomial.Xstatement and proof · cited by 1,639
pow_oneproof · cited by 894

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Polynomial.Chebyshev.one_sub_X_sq_mul_iterate_derivative_T_evalproof · cited by 2