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Formula for the Legendre polynomials From Wikipedia, the free encyclopedia

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.

Let be a sequence of orthogonal polynomials defined on the interval satisfying the orthogonality condition
where is a suitable weight function, is a constant depending on , and is the Kronecker delta. If the weight function satisfies the following differential equation (called Pearson's 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 relation of the form,
for some constants . This relation is called *Rodrigues' type formula*, or just *Rodrigues' formula*.^{[1]}

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 *L*_{0}, *L*_{1}, ..., 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.

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