Rodrigues' formula

In mathematics, Rodrigues' formula generates the Legendre polynomials. It was independently introduced by, and. The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. describes the history of the Rodrigues formula in detail.

Statement

Let be a sequence of orthogonal polynomials on the interval with respect to weight function. That is, they have degrees, satisfy the orthogonality condition
where are nonzero constants depending on, and is the Kronecker delta. The interval may be infinite in one or both ends.
More abstractly, this can be viewed through Sturm–Liouville theory. Define an operator, then the differential equation is equivalent to. Define the functional space as the Hilbert space of functions over, such that. Then the operator is self-adjoint on functions satisfying certain boundary conditions, allowing us to apply the spectral theorem.

Generating function

A simple argument using Cauchy's integral formula shows that the orthogonal polynomials obtained from the Rodrigues formula have a generating function of the form
The functions here may not have the standard normalizations. But we can write this equivalently as
where the are chosen according to the application so as to give the desired normalizations. The variable u may be replaced by a constant multiple of u so that
This gives an alternate form of the generating function.
By Cauchy's integral formula, Rodrigues’ formula is equivalent towhere the integral is along a counterclockwise closed loop around. Let
Then the complex path integral takes the form
where now the closed path C encircles the origin. In the equation for, is an implicit function of. Expanding in the power series given earlier gives
Only the term has a nonzero residue, which is. The coefficient was dropped since normalizations are conventions which can be inserted afterwards as discussed earlier.
By expressing t in terms of u in the general formula just given for, explicit formulas for may be found. As a simple example, let and so that,, and so.

Examples

These formulae
are for the classical orthogonal polynomials. Similar formulae hold for many other sequences of orthogonal functions arising from Sturm–Liouville equations, and these are also called the Rodrigues formula, especially when the resulting sequence is polynomial.

Legendre

Source:
Rodrigues stated his formula for Legendre polynomials :
For Legendre polynomials, the generating function is defined as
The contour integral gives the Schläfli integral for Legendre polynomials:
Summing up the integrand
where. For small, we have, which heuristically suggests that the integral should be the residue around, thus giving

Hermite

Source:
Physicist's Hermite polynomials:
The generating function is defined as
The contour integral gives

Laguerre

Source:
For associated Laguerre polynomials
The generating function is defined as
By the same method, we have.

Jacobi

Source:
where, and the branch of square root is chosen so that.