In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) generates the Legendre polynomials.
It was independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827).
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.
Askey (2005) describes the history of the Rodrigues formula in detail.
be a sequence of orthogonal polynomials defined on the interval
satisfying the orthogonality condition
is a suitable weight function,
The weight function w(x)=W(x)/B(x) where the integration factor W(x) satisfies the equation
For instance, for the Legendre polynomials, B(x)=1-x*x and A(x)=-2x.
satisfies a relation of the form,
This relation is called Rodrigues' type formula, or just Rodrigues' formula.
[1] Polynomials obtained from Rodrigues' formula obey the second order differential equation for the classical orthogonal polynomials
Continuing the example of the Legendre polynomials,
The following proof shows that the polynomials obtained from the Rodrigues' formula obey the second order differential equation just given.
the differential equation that we are to prove may be put in the form
This is the differential equation that we will prove to be true.
Rodrigues’ formula together with Cauchy’s Residue theorem for complex integration on a closed path enclosing poles gives the generating functions having the property
Here's how: By Cauchy’s Residue Theorem, Rodrigues’ formula is equivalent to
where the complex variable t is integrated along a counterclockwise closed path C that encircles x.
Then the complex path integral takes the form
where now the closed path C encircles the origin.
As examples, we will find the generating functions for the Hermite polynomials and the Legendre polynomials.
, we get the usual generating function relationship.
The Legendre polynomials require more work.
, we get the generating function relation in the usual form.
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 polynomials 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 (or Rodrigues' type formula) for that case, especially when the resulting sequence is polynomial.