Theorems · Theorem · commutative algebra
Polynomial.factorial_mul_shiftedLegendre_eq
∀ (n : ℕ), ↑n.factorial * Polynomial.shiftedLegendre n = (⇑Polynomial.derivative)^[n] (Polynomial.X ^ n * (1 - Polynomial.X) ^ n)
The shifted Legendre polynomial multiplied by a factorial equals the higher-order derivative of
the combinatorial function X ^ n * (1 - X) ^ n. This is the analogue of Rodrigues' formula for
the shifted Legendre polynomials.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites47
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- Polynomial.Xstatement and proof · cited by 1,639
- Polynomial.Cproof · cited by 1,598
- add_commproof · cited by 1,535
Cited by1
Results whose statement or proof uses this declaration.
- Polynomial.neg_one_pow_mul_shiftedLegendre_comp_one_sub_X_eqproof · cited by 1