`BookProof.ChapterLegendrePolynomial.legendre_deriv_coeff_rec` (l μ j : ℕ) : (((j : ℝ) + 2) * ((j : ℝ) + 1)) * (derivative^[μ] (legendre l)).coeff (j + 2) = ((j : ℝ) * ((j : ℝ) - 1

Proved

BookProof.ChapterLegendrePolynomial.legendre_deriv_coeff_rec

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

theoremstimepiece

Prove the following Lean 4 theorem from ChapterLegendrePolynomial.

BookProof.ChapterLegendrePolynomial.legendre_deriv_coeff_rec (l μ j : ℕ) : (((j : ℝ) + 2) * ((j : ℝ) + 1)) * (derivative^[μ] (legendre l)).coeff (j + 2) = ((j : ℝ) * ((j : ℝ) - 1) + (2 * (μ : ℝ) + 2) * j + ((μ : ℝ) * ((μ : ℝ) + 1) - (l : ℝ) * ((l : ℝ) + 1))) * (derivative^[μ] (legendre l)).coeff j

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

Preamble

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



open Polynomial

Formal statement

theorem BookProof.ChapterLegendrePolynomial.legendre_deriv_coeff_rec (l μ j : ℕ) :
    (((j : ℝ) + 2) * ((j : ℝ) + 1)) * (derivative^[μ] (legendre l)).coeff (j + 2)
      = ((j : ℝ) * ((j : ℝ) - 1) + (2 * (μ : ℝ) + 2) * j
          + ((μ : ℝ) * ((μ : ℝ) + 1) - (l : ℝ) * ((l : ℝ) + 1)))
        * (derivative^[μ] (legendre l)).coeff j := by sorry

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