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
Gegenbauer polynomials with α=1
Gegenbauer polynomials with α=2
Gegenbauer polynomials with α=3
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.
Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (
Suetin 2001):
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.[1]
For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun
p. 774)
To wit, for n ≠ m,
They are normalized by
Applications
The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of
potential theory and
harmonic analysis. The
Newtonian potential in Rn has the expansion, valid with α = (n − 2)/2,
It follows that the quantities are
spherical harmonics, when regarded as a function of x only. They are, in fact, exactly the
zonal spherical harmonics, up to a normalizing constant.
In
spectral methods for solving differential equations, if a function is expanded in the basis of
Chebyshev polynomials and its derivative is represented in a Gegenbauer/ultraspherical basis, then the derivative operator becomes a
diagonal matrix, leading to fast
banded matrix methods for large problems.[2]
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
Gegenbauer polynomials with α=1
Gegenbauer polynomials with α=2
Gegenbauer polynomials with α=3
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.
Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (
Suetin 2001):
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.[1]
For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun
p. 774)
To wit, for n ≠ m,
They are normalized by
Applications
The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of
potential theory and
harmonic analysis. The
Newtonian potential in Rn has the expansion, valid with α = (n − 2)/2,
It follows that the quantities are
spherical harmonics, when regarded as a function of x only. They are, in fact, exactly the
zonal spherical harmonics, up to a normalizing constant.
In
spectral methods for solving differential equations, if a function is expanded in the basis of
Chebyshev polynomials and its derivative is represented in a Gegenbauer/ultraspherical basis, then the derivative operator becomes a
diagonal matrix, leading to fast
banded matrix methods for large problems.[2]