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Concept
Rodrigues' formula
Summary
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi 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.
Let be a sequence of orthogonal polynomials 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 recurrence relation of the form,
for some 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 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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Ontological neighbourhood
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Related concepts (4)
Orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases.
Classical orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials (including as a special case the Gegenbauer polynomials, Chebyshev polynomials, and Legendre polynomials). They have many important applications in such areas as mathematical physics (in particular, the theory of random matrices), approximation theory, numerical analysis, and many others.
Legendre polynomials
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.