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In [[mathematics]], '''Gegenbauer polynomials''' or '''ultraspherical polynomials''' ''C''{{su|p=(α)|b=''n''}}(''x'') are  [[orthogonal polynomials]] on the interval [&minus;1,1] with respect to the [[weight function]] (1&nbsp;&minus;&nbsp;''x''<sup>2</sup>)<sup>''α''–1/2</sup>. They generalize [[Legendre polynomials]] and [[Chebyshev polynomials]], and are special cases of [[Jacobi polynomials]]. They are named after [[Leopold Gegenbauer]].
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==Characterizations==
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A variety of characterizations of the Gegenbauer polynomials are available.
 
* The polynomials can be defined in terms of their [[generating function]] {{harv|Stein|Weiss|1971|loc=§IV.2}}:
 
::<math>\frac{1}{(1-2xt+t^2)^\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x) t^n.</math>
 
* The polynomials satisfy the [[recurrence relation]] {{harv|Suetin|2001}}:
 
::<math>
\begin{align}
C_0^\alpha(x) & = 1 \\
C_1^\alpha(x) & = 2 \alpha x \\
C_n^\alpha(x) & = \frac{1}{n}[2x(n+\alpha-1)C_{n-1}^\alpha(x) - (n+2\alpha-2)C_{n-2}^\alpha(x)].
\end{align}
</math>
 
* Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation {{harv|Suetin|2001}}:
 
::<math>(1-x^{2})y''-(2\alpha+1)xy'+n(n+2\alpha)y=0.\,</math>
 
:When ''&alpha;''&nbsp;=&nbsp;1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the [[Legendre polynomials]].
 
* They are given as [[Gaussian hypergeometric series]] in certain cases where the series is in fact finite:
 
::<math>C_n^{(\alpha)}(z)=\frac{(2\alpha)_n}{n!}
\,_2F_1\left(-n,2\alpha+n;\alpha+\frac{1}{2};\frac{1-z}{2}\right).</math>
 
:(Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_561.htm p.&nbsp;561]). Here (2&alpha;)<sub>''n''</sub> is the [[rising factorial]].   Explicitly,
::<math>
C_n^{(\alpha)}(z)=\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)k!(n-2k)!}(2z)^{n-2k}.
</math>
* They are special cases of the [[Jacobi polynomials]] {{harv|Suetin|2001}}:
::<math>C_n^{(\alpha)}(x) = \frac{(2\alpha)_n}{(\alpha+\frac{1}{2})_{n}}P_n^{(\alpha-1/2,\alpha-1/2)}(x).</math>
:in which <math>(\theta)_n</math> represents the [[rising factorial]] of <math>\theta</math>.  
:One therefore also has the [[Rodrigues formula]]
::<math>C_n^{(\alpha)}(x) = \frac{(-2)^n}{n!}\frac{\Gamma(n+\alpha)\Gamma(n+2\alpha)}{\Gamma(\alpha)\Gamma(2n+2\alpha)}(1-x^2)^{-\alpha+1/2}\frac{d^n}{dx^n}\left[(1-x^2)^{n+\alpha-1/2}\right].</math>
 
==Orthogonality and normalization==
 
For a fixed ''α'', the polynomials are orthogonal on [&minus;1,&nbsp;1] with respect to the weighting function (Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_774.htm p.&nbsp;774])
 
:<math> w(z) = \left(1-z^2\right)^{\alpha-\frac{1}{2}}.</math>
 
To wit, for ''n''&nbsp;≠&nbsp;''m'',
 
:<math>\int_{-1}^1 C_n^{(\alpha)}(x)C_m^{(\alpha)}(x)(1-x^2)^{\alpha-\frac{1}{2}}\,dx = 0.</math>
 
They are normalized by
 
:<math>\int_{-1}^1 \left[C_n^{(\alpha)}(x)\right]^2(1-x^2)^{\alpha-\frac{1}{2}}\,dx = \frac{\pi 2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)[\Gamma(\alpha)]^2}.</math>
 
==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'''<sup>''n''</sup> has the expansion, valid with α&nbsp;=&nbsp;(''n''&nbsp;&minus;&nbsp;2)/2,
 
:<math>\frac{1}{|\mathbf{x}-\mathbf{y}|^{n-2}} = \sum_{k=0}^\infty \frac{|\mathbf{x}|^k}{|\mathbf{y}|^{k+n-2}}C_{n,k}^{(\alpha)}(\mathbf{x}\cdot \mathbf{y}).</math>
 
When ''n''&nbsp;=&nbsp;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 {{harv|Stein|Weiss|1971}}.
 
It follows that the quantities <math>C^{((n-2)/2)}_{n,k}(\mathbf{x}\cdot\mathbf{y})</math> are [[spherical harmonics]], when regarded as a function of '''x''' only. They are, in fact, exactly the [[zonal spherical harmonic]]s, up to a normalizing constant.
 
Gegenbauer polynomials also appear in the theory of [[Positive-definite function]]s.
 
The [[Askey–Gasper inequality]] reads
:<math>\sum_{j=0}^n\frac{C_j^\alpha(x)}{{2\alpha+j-1\choose j}}\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).</math>
 
== See also==
 
* [[Rogers polynomials]], the ''q''-analogue of Gegenbauer polynomials
* [[Chebyshev polynomials]]
 
== References ==
 
* {{Abramowitz_Stegun_ref|22|773}}
* {{citation|last=Bayin|first=S.S.|year=2006|title=Mathematical Methods in Science and Engineering|publisher=Wiley}}, Chapter 5.
*{{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}
* {{citation|last1=Stein|first1=Elias|authorlink1=Elias Stein|first2=Guido|last2=Weiss|authorlink2=Guido Weiss|title=Introduction to Fourier Analysis on Euclidean Spaces|publisher=Princeton University Press|year=1971|isbn=978-0-691-08078-9|location=Princeton, N.J.}}.
* {{springer|title=Ultraspherical polynomials|id=U/u095030|first=P.K.|last=Suetin}}.
 
[[Category:Orthogonal polynomials]]
[[Category:Special hypergeometric functions]]

Revision as of 13:25, 28 February 2014

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