Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Polynôme de Gegenbauer
Résumé
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.
File:Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
File:Mplwp gegenbauer Cn05a1.svg|Gegenbauer polynomials with ''α''=1
File:Mplwp gegenbauer Cn05a2.svg|Gegenbauer polynomials with ''α''=2
File:Mplwp gegenbauer Cn05a3.svg|Gegenbauer polynomials with ''α''=3
File:Gegenbauer polynomials.gif|An animation showing the polynomials on the ''xα''-plane for the first 4 values of ''n''.
A variety of characterizations of the Gegenbauer polynomials are available.
The polynomials can be defined in terms of their generating function :
The polynomials satisfy the recurrence relation :
Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation :
When α = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials.
When α = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the Chebyshev polynomials of the second kind.
They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:
(Abramowitz & Stegun p. 561). Here (2α)n is the rising factorial. Explicitly,
From this it is also easy to obtain the value at unit argument:
They are special cases of the Jacobi polynomials :
in which represents the rising factorial of .
One therefore also has the Rodrigues formula
For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun p.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Proximité ontologique