Gegenbauer polynomials

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In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(α)
n
(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − 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[edit]

A variety of characterizations of the Gegenbauer polynomials are available.

\frac{1}{(1-2xt+t^2)^\alpha}=\sum_{n=0}^\infty C_n^{(\alpha)}(x) t^n.

\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}
  • Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):
(1-x^{2})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.
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 (2α)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}.
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{(-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].

Orthogonality and normalization[edit]

For a fixed α, the polynomials are orthogonal on [−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[edit]

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,

\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}).

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 (Stein & Weiss 1971).

It follows that the quantities C^{((n-2)/2)}_{n,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)}{{2\alpha+j-1\choose j}}\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).

See also[edit]

References[edit]