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Concept
Gegenbauer polynomials
Summary
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.
Official source
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