Ultraspherical polynomial

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.Arfken, Weber, and Harris (2013) "Mathematical Methods for Physicists", 7th edition; ch. 18.4

* 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,
::$C_n^(z)=\sum_^ (-1)^k\frac(2z)^.$
* 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

Orthogonality and normalization

For a fixed ''α'', 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.

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