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

Askey, Richard (2005), "The 1839 paper on permutations: its relation to the Rodrigues formula and further developments", in Altmann, Simón L.; Ortiz, Eduardo L., Mathematics and social utopias in France: Olinde Rodrigues and his times, History of mathematics, 28, Providence, R.I.: American Mathematical Society, pp. 105–118, ISBN 978-0-8218-3860-0
Ivory, James (1824), "On the Figure Requisite to Maintain the Equilibrium of a Homogeneous Fluid Mass That Revolves Upon an Axis", Philosophical Transactions of the Royal Society of London, The Royal Society, 114: 85–150, doi:10.1098/rstl.1824.0008, JSTOR 107707
Jacobi, C. G. J. (1827), "Ueber eine besondere Gattung algebraischer Functionen, die aus der Entwicklung der Function (1 − 2xz + z²)^1/2 entstehen.", Journal für Reine und Angewandte Mathematik (in German), 2: 223–226, doi:10.1515/crll.1827.2.223, ISSN 0075-4102
O'Connor, John J.; Robertson, Edmund F., "Olinde Rodrigues", MacTutor History of Mathematics archive, University of St Andrews .
Rodrigues, Olinde (1816), "De l'attraction des sphéroïdes", Correspondence sur l'École Impériale Polytechnique, (Thesis for the Faculty of Science of the University of Paris), 3 (3): 361–385

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