Rodrigues' Formula: Building Orthogonal Polynomials by Repeated Differentiation

Precise statement

Let w > 0 be a weight on an interval (a,b) ⊂ ℝ with all moments finite, and let s be a polynomial with deg s ≤ 2 that together with w satisfies the Pearson condition: (w s)′ = w · τ for some polynomial τ with deg τ = 1. Suppose the boundary terms vanish: w(x) sⁿ(x) x^k → 0 as x → a⁺, b⁻ for all n, k ≥ 0. Define

Rₙ(x) = (1/(kₙ w(x))) · (dⁿ/dxⁿ)[ w(x) s(x)ⁿ ],

with kₙ a normalizing constant. Claim: Rₙ is a polynomial of degree exactly n, and ⟨Rₘ, Rₙ⟩_w := ∫ₐᵇ Rₘ Rₙ w dx = 0 for m ≠ n. Thus {Rₙ}ₙ≥₀ is the sequence of orthogonal polynomials for the inner product ⟨·,·⟩_w. For Legendre, w = 1, s = x²−1, kₙ = 2ⁿ n!; the classical form Pₙ(x) = (1/(2ⁿn!)) dⁿ/dxⁿ (x²−1)ⁿ is exactly this.

The picture

Why should differentiating a weight n times manufacture orthogonality? Think of dⁿ/dxⁿ as an operator that is adjoint to multiplication by low-degree polynomials under integration by parts. Rₙ is built so that w·sⁿ is an nth antiderivative: it, and its first n−1 derivatives, all carry a factor of s that vanishes at the endpoints (for Legendre, s = x²−1 is zero at ±1). So when you integrate Rₙ against any polynomial of degree < n, you can integrate by parts n times, throwing derivatives off the polynomial. After n integrations a degree-(

Key idea of the proof

The mechanism is a single lemma: Rₙ is orthogonal to every polynomial q of degree < n. Compute ∫ₐᵇ q(x) Rₙ(x) w(x) dx = (1/kₙ) ∫ₐᵇ q · (w sⁿ)⁽ⁿ⁾ dx. Integrate by parts n times. Each boundary term is [q⁽ʲ⁾ · (w sⁿ)⁽ⁿ⁻¹⁻ʲ⁾]ₐᵇ; but (w sⁿ)⁽ᵏ⁾ for k < n still contains a factor of s (this is where deg s ≤ 2 and the Pearson relation are used to keep degrees controlled), and s·w → 0 at a,b, so every boundary term vanishes. After n steps the integral becomes (±1/kₙ) ∫ₐᵇ q⁽ⁿ⁾ · (w sⁿ) dx. Since deg q < n, q⁽ⁿ⁾ ≡ 0, and the integral is 0. That Rₙ has degree exactly n follows from the Pearson relation, which forces (w sⁿ)⁽ⁿ⁾/w to be a degree-n polynomial with nonzero leading coefficient ∏ⱼ₌₀ⁿ⁻¹(τ′·? ) — concretely a product of the linear coefficients that never degenerates. Distinct-degree orthogonal polynomials are automatically mutually orthogonal, so we are done.

Worked example: Legendre P₂

Take w = 1, s = x²−1, n = 2. Then s² = (x²−1)² = x⁴ − 2x² + 1. Differentiate twice: (d/dx)(x⁴−2x²+1) = 4x³ − 4x, and (d/dx)(4x³−4x) = 12x² − 4. Divide by k₂ = 2²·2! = 8:

P₂(x) = (12x² − 4)/8 = (3x² − 1)/2.

Check orthogonality to 1 and x on [−1,1]: ∫₋₁¹ (3x²−1)/2 dx = ½[x³ − x]₋₁¹ = ½[(1−1) − (−1+1)] = 0, and ∫₋₁¹ x·(3x²−1)/2 dx = 0 by oddness. Both vanish, confirming P₂ ⊥ span{1, x}. And P₂(1) = (3−1)/2 = 1, the standard normalization Pₙ(1) = 1 that Rodrigues' constant 2ⁿn! delivers automatically. One more derivative-and-divide gives P₃ = (5x³ − 3x)/2 — no Gram–Schmidt, no recursion needed.

Why the hypotheses matter

The Pearson relation (w s)′ = w·τ with deg s ≤ 2, deg τ = 1 is not decoration — it is what pins down the only weights that give a Rodrigues formula. Bochner's theorem (1929) proves that {Legendre, Hermite, Laguerre, Jacobi} (with their affine images) are the only orthogonal polynomial families that are also eigenfunctions of a second-order differential operator, equivalently the only ones with a classical Rodrigues formula. Drop the boundary-vanishing condition and the integration-by-parts boundary terms survive, destroying orthogonality. Drop deg s ≤ 2 and (w sⁿ)⁽ⁿ⁾/w need not be a polynomial of degree n — degree control fails. A weight like w = e^(−x⁴) on ℝ generates perfectly good orthogonal polynomials by Gram–Schmidt, yet has no finite Rodrigues formula: (w s)′/w cannot be linear for any polynomial s. Rodrigues is thus a rigidity phenomenon, closely tied to the Sturm–Liouville theory (self-adjoint operator L with L Rₙ = λₙ Rₙ) behind these polynomials.

Why it matters

Rodrigues' formula is the computational and structural engine of the classical special functions. It gives instant proofs of the three-term recurrence, of generating functions (e.g. 1/√(1−2xt+t²) = ∑ Pₙ(x)tⁿ), and of the differential equations these polynomials solve — the Legendre equation appears the moment you separate variables in the Laplacian on the sphere, so Pₙ(cos θ) are the zonal spherical harmonics that underlie multipole expansions, gravitational potentials, and quantum angular momentum. Hermite polynomials from the Rodrigues form eᵗ(−1)ⁿe^(x²)(d/dx)ⁿe^(−x²) are the quantum harmonic oscillator eigenstates; Laguerre polynomials are the radial hydrogen wavefunctions. In numerical analysis the roots of these polynomials are the nodes of Gauss quadrature, exact for polynomials up to degree 2n−1. The formula also generalizes: matrix-valued and multivariate Rodrigues formulas power the theory of orthogonal polynomials on symmetric spaces.

The three classical families produced by Rodrigues' formula, with their weight w, Pearson polynomial s, interval, and constant kₙ.

Family	Weight w(x)	s(x) / interval	Rodrigues formula
Legendre Pₙ	1	(x²−1) on [−1,1]	(1/(2ⁿn!)) (d/dx)ⁿ(x²−1)ⁿ
Hermite Hₙ (physicists)	e^(−x²)	1 on ℝ	(−1)ⁿ e^(x²) (d/dx)ⁿ e^(−x²)
Laguerre Lₙ^(α)	xᵅe^(−x)	x on [0,∞)	(x^(−α)e^x/n!) (d/dx)ⁿ(x^(n+α)e^(−x))
Jacobi Pₙ^(α,β)	(1−x)ᵅ(1+x)ᵝ	(x²−1) on [−1,1]	((−1)ⁿ/(2ⁿn!)) w^(−1)(d/dx)ⁿ[w(x²−1)ⁿ]
Chebyshev Tₙ	(1−x²)^(−1/2)	(x²−1) on [−1,1]	Gegenbauer/Jacobi special case (α=β=−½)