Rodrigues' formula

For the 3-dimensional rotation formula, see Rodrigues' rotation formula.

In mathematics, Rodrigues's formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials 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.

Statement

Rodrigues stated his formula for Legendre polynomials $P_n$:

$P_n(x) = {1 \over 2^n n!} {d^n \over dx^n } \left[ (x^2 -1)^n \right].$

Laguerre polynomials are usually denoted L₀, L₁, ..., and the Rodrigues formula can be written as

$L_n(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}\left(e^{-x} x^n\right) =\frac{1}{n!} \left( \frac{d}{dx} -1 \right) ^n x^n,$

The Rodrigues formula for the Hermite polynomial can be written as

$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2}=\left (2x-\frac{d}{dx} \right )^n \cdot 1$.

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.

References

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