Gegenbauer polynomials

In mathematics, Gegenbauer polynomials or ultraspherical polynomials C$(α) n$(x) are orthogonal polynomials on the interval [&minus;1,1] with respect to the weight function (1 &minus; x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.

Characterizations
A variety of characterizations of the Gegenbauer polynomials are available.


 * The polynomials can be defined in terms of their generating function :


 * $$\frac{1}{(1-2xt+t^2)^\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x) t^n \qquad (0 \leq |x| < 1, |t| \leq 1, \alpha > 0)$$


 * The polynomials satisfy the recurrence relation :



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


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


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


 * When &alpha; = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials.
 * When &alpha; = 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^{(\alpha)}(z)=\frac{(2\alpha)_n}{n!}

\,_2F_1\left(-n,2\alpha+n;\alpha+\frac{1}{2};\frac{1-z}{2}\right).$$


 * (Abramowitz & Stegun p. 561). Here (2&alpha;)n is the rising factorial.  Explicitly,

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}. $$
 * From this it is also easy to obtain the value at unit argument:

C_n^{(\alpha)}(1)=\frac{\Gamma(2\alpha+n)}{\Gamma(2\alpha)n!}. $$
 * They are special cases of the Jacobi polynomials :
 * $$C_n^{(\alpha)}(x) = \frac{(2\alpha)_n}{(\alpha+\frac{1}{2})_{n}}P_n^{(\alpha-1/2,\alpha-1/2)}(x).$$
 * in which $$(\theta)_n$$ represents the rising factorial of $$\theta$$.
 * One therefore also has the Rodrigues formula
 * $$C_n^{(\alpha)}(x) = \frac{(-1)^n}{2^n n!}\frac{\Gamma(\alpha+\frac{1}{2})\Gamma(n+2\alpha)}{\Gamma(2\alpha)\Gamma(\alpha+n+\frac{1}{2})}(1-x^2)^{-\alpha+1/2}\frac{d^n}{dx^n}\left[(1-x^2)^{n+\alpha-1/2}\right].$$

Orthogonality and normalization
For a fixed α > -1/2, the polynomials are orthogonal on [&minus;1, 1] with respect to the weighting function (Abramowitz & Stegun p. 774)


 * $$ w(z) = \left(1-z^2\right)^{\alpha-\frac{1}{2}}.$$

To wit, for n ≠ m,


 * $$\int_{-1}^1 C_n^{(\alpha)}(x)C_m^{(\alpha)}(x)(1-x^2)^{\alpha-\frac{1}{2}}\,dx = 0.$$

They are normalized by


 * $$\int_{-1}^1 \left[C_n^{(\alpha)}(x)\right]^2(1-x^2)^{\alpha-\frac{1}{2}}\,dx = \frac{\pi 2^{1-2\alpha}\Gamma(n+2\alpha)}{n!(n+\alpha)[\Gamma(\alpha)]^2}.$$

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 &minus; 2)/2,


 * $$\frac{1}{|\mathbf{x}-\mathbf{y}|^{n-2}} = \sum_{k=0}^\infty \frac{|\mathbf{x}|^k}{|\mathbf{y}|^{k+n-2}}C_k^{(\alpha)}(\frac{\mathbf{x}\cdot \mathbf{y}}{|\mathbf{x}||\mathbf{y}|}).$$

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 $$C^{((n-2)/2)}_k(\mathbf{x}\cdot\mathbf{y})$$ 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
 * $$\sum_{j=0}^n\frac{C_j^\alpha(x)}\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).$$

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.