Gegenbauer Polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials ''C''(''x'') are orthogonal polynomials on the interval minus;1,1with respect to the weight function (1 − ''x''²)^''α''–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.

Characterizations

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 :

::$\frac=\sum_^\infty C_n^(x) t^n \qquad (0 \leq , x, < 1, , t, \leq 1, \alpha > 0)$

* The polynomials satisfy the recurrence relation :

::$\begin C_0^(x) & = 1 \\ C_1^(x) & = 2 \alpha x \\ (n+1) C_^(x) & = 2(n+\alpha) x C_^(x) - (n+2\alpha-1)C_^(x). \end$

* Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation :

::$(1-x^)y''-(2\alpha+1)xy'+n(n+2\alpha)y=0.\,$

: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:

::$C_n^(z)=\frac \,_2F_1\left(-n,2\alpha+n;\alpha+\frac;\frac\right).$

:(Abramowitz & Stegu
p. 561
. Here (2α)_''n'' is the rising factorial. Explicitly,
::$C_n^(z)=\sum_^ (-1)^k\frac(2z)^.$
:From this it is also easy to obtain the value at unit argument:
::$C_n^(1)=\frac.$
* They are special cases of the Jacobi polynomials :
::$C_n^(x) = \fracP_n^(x).$
:in which $(\theta)_n$ represents the rising factorial of $\theta$.
:One therefore also has the Rodrigues formula
::C_n^(x) = \frac\frac(1-x^2)^\frac\left 1-x^2)^\right

* An alternative normalization sets $C_n^(1)=1$. Assuming this alternative normalization, the derivatives of Gegenbauer are expressed in terms of Gegenbauer:
$\begin \fracC_^(x)=\frac & \sum_^j \frac \\ & \times \frac(q+j-i-1)!C_^(x) \end$

Orthogonality and normalization

For a fixed ''α > -1/2'', the polynomials are orthogonal on minus;1, 1with respect to the weighting function (Abramowitz & Stegu
p. 774

:$w(z) = \left(1-z^2\right)^.$

To wit, for ''n'' ≠ ''m'',

:$\int_^1 C_n^(x)C_m^(x)(1-x^2)^\,dx = 0.$

They are normalized by

:\int_^1 \left _n^(x)\right2(1-x^2)^\,dx = \frac.

Applications

The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potential theory and harmonic analysis. The Newtonian potential in R^''n'' has the expansion, valid with α = (''n'' − 2)/2,

:$\frac = \sum_^\infty \fracC_k^(\frac).$

When ''n'' = 3, this gives the Legendre polynomial expansion of the gravitational potential. Similar expressions are available for the expansion of the Poisson kernel in a ball .

It follows that the quantities $C^_k(\mathbf\cdot\mathbf)$ 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.

Gegenbauer polynomials also appear in the theory of positive-definite functions.

The Askey–Gasper inequality reads
:$\sum_^n\frac\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).$

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.

Other properties

Dirichlet–Mehler-type integral representation:$\frac=\frac=\frac(\sin\theta)^\int_^\frac\,\mathrm\phi,$Laplace-type$\begin \frac & =\frac \\ & =\frac \int_0^\pi(\cos \theta+i \sin \theta \cos \phi)^n(\sin \phi)^ \mathrm \phi \end$

See also

* Rogers polynomials, the ''q''-analogue of Gegenbauer polynomials
* Chebyshev polynomials
* Romanovski polynomials

References

* *
* .
* {{springer, title=Ultraspherical polynomials, id=U/u095030, first=P.K., last=Suetin.

;Specific

Orthogonal polynomials
Special hypergeometric functions