`BookProof.ChapterLegendrePolynomial.iterD_Xsq_mul` (k : ℕ) (f : ℝ[X]) : derivative^[k] (X ^ 2 * f) = X ^ 2 * derivative^[k] f + C (2 * (k : ℝ)) * X * derivative^[k-1] f + C ((k :

Proved

BookProof.ChapterLegendrePolynomial.iterD_Xsq_mul

by leonardopedro · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLegendrePolynomial.

BookProof.ChapterLegendrePolynomial.iterD_Xsq_mul (k : ℕ) (f : ℝ[X]) : derivative^[k] (X ^ 2 * f) = X ^ 2 * derivative^[k] f + C (2 * (k : ℝ)) * X * derivative^[k-1] f + C ((k : ℝ) * ((k : ℝ) - 1)) * derivative^[k-2] f

Formalization note: Lean 4 identifier BookProof.ChapterLegendrePolynomial.iterD_Xsq_mul.

Preamble

-- Generated from ChapterLegendrePolynomial.lean — theorem BookProof.ChapterLegendrePolynomial.iterD_Xsq_mul
import Mathlib
import Definitions.Def_ChapterLegendrePolynomial
open BookProof.ChapterLegendrePolynomial



open Polynomial

Formal statement

theorem BookProof.ChapterLegendrePolynomial.iterD_Xsq_mul (k : ℕ) (f : ℝ[X]) :
    derivative^[k] (X ^ 2 * f)
      = X ^ 2 * derivative^[k] f + C (2 * (k : ℝ)) * X * derivative^[k-1] f
        + C ((k : ℝ) * ((k : ℝ) - 1)) * derivative^[k-2] f := by sorry

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