Rodrigues' formula


In mathematics, Rodrigues' formula is a formula for the Legendre polynomials 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 satisfying the orthogonality condition
where, is a suitable weight function, are constants and is the Kronecker delta. If the weight function satisfies the following differential equation,
where is a polynomial with degree at most 1 and is a polynomial with degree at most 2 and, further, the limits
then, it can be shown that satisfies a recurrence relation of the form,
for a given constants. This relation is called Rodrigues' type formula, or just Rodrigues' formula.
The most known applications of Rodrigues' type formulas are the formulas for Legendre, Laguerre and Hermite polynomials:
Rodrigues stated his formula for Legendre polynomials :
Laguerre polynomials are usually denoted L0, L1, ..., and the Rodrigues formula can be written as
The Rodrigues formula for the Hermite polynomial can be written as
Similar formulae hold for many other sequences of orthogonal functions arising from Sturm-Liouville equations, and these are also called the Rodrigues formula for that case, especially when the resulting sequence is polynomial.