Rodrigues' formula
/ Rodrigues: ro-DREE-gez /
There are several ways to get hold of the classical orthogonal polynomials — solve the equation by series, expand a generating function, or run a recurrence. Rodrigues' formula offers a fourth, often the slickest: it hands you the whole polynomial in closed form as a high derivative of a simple expression. No series, no recursion — just differentiate.
The pattern is uniform across the families. The n-th polynomial p_n(x) equals a constant times one over a weight function w(x), times the n-th derivative of (w(x) times a fixed function raised to the n-th power). For the Legendre polynomials it is the elegant P_n(x) = 1/(2^n n!) times d^n/dx^n of (x^2 - 1)^n. For the Hermite polynomials it is H_n(x) = (-1)^n e^{x^2} times d^n/dx^n of e^{-x^2}; for Laguerre, L_n(x) = (e^x / n!) times d^n/dx^n of (x^n e^{-x}). In each case you take a smooth seed, differentiate n times, and a degree-n polynomial pops out.
Rodrigues' formula is more than a computational shortcut — it makes the deepest property, orthogonality, almost obvious. Because the polynomial is an n-fold derivative, integrating p_n against any lower-degree polynomial and repeatedly integrating by parts moves all n derivatives onto the lower-degree factor, which differentiates to zero after fewer than n steps; the integral vanishes. So the formula gives a one-line proof that p_n is orthogonal to every polynomial of lower degree under the weight w. It is the single cleanest bridge from a differential equation to an orthogonal family.
Compute P_2 by Rodrigues: take (x^2 - 1)^2 = x^4 - 2x^2 + 1, differentiate twice to get 12x^2 - 4, multiply by 1/(2^2 times 2!) = 1/8, giving (12x^2 - 4)/8 = (3x^2 - 1)/2 = P_2(x). The same recipe with n derivatives gives every Legendre polynomial.
Differentiate (x^2 - 1)^n n times and normalise — out comes P_n.
Rodrigues' formula in this clean 'n-th derivative' shape exists only for the classical families (Legendre, Hermite, Laguerre, and a generalised version for Jacobi). It is not a universal trick for every set of orthogonal polynomials, and the normalising constant differs family to family.