In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − 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.
Characterizations
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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.
</gallery>
A variety of characterizations of the Gegenbauer polynomials are available.
- The polynomials can be defined in terms of their generating function:
::<math>\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)</math>
- The polynomials satisfy the recurrence relation:
::<math>
\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}
</math>
- Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation:
- 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>
: Here (2α)<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>
:From this it is also easy to obtain the value at unit argument:
::<math>
C_n^{(\alpha)}(1)=\frac{\Gamma(2\alpha+n)}{\Gamma(2\alpha)n!}.
</math>
- They are special cases of the Jacobi polynomials:
<math display="block">\begin{aligned}
\frac{d^q}{dx^q}C_{q+2 j+1}^{(\alpha)}(x)=\frac{2^q(q+2 j+1)!}{(q-1)!\Gamma(q+2 j+2 \alpha+1)} & \sum_{i=0}^j \frac{(2 i+\alpha+1) \Gamma(2 i+2 \alpha+1)}{(2 i+1)!(j-i)!} \\
& \times \frac{\Gamma(q+j+i+\alpha+1)}{\Gamma(j+i+\alpha+2)}(q+j-i-1)!C_{2 i+1}^{(\alpha)}(x)
\end{aligned}</math>
Orthogonality and normalization
For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function
:<math> w(z) = \left(1-z^2\right)^{\alpha-\frac{1}{2.</math>
To wit, for n ≠ 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 = (n − 2)/2,
:<math>\frac{1}{|\mathbf{x}-\mathbf{y}|^{n-2 = \sum_{k=0}^\infty \frac{|\mathbf{x}|^k}{|\mathbf{y}|^{k+n-2C_k^{(\alpha)}(\frac{\mathbf{x}\cdot \mathbf{y{|\mathbf{x}||\mathbf{y}|}).</math>
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 <math>C^{((n-2)/2)}_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 harmonics, up to a normalizing constant.
Gegenbauer polynomials also appear in the theory of positive-definite functions.
The Askey–Gasper inequality reads
:<math>\sum_{j=0}^n\frac{C_j^\alpha(x)}\ge 0\qquad (x\ge-1,\, \alpha\ge 1/4).</math>
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:<math display="block">\frac{P^{(\alpha,\alpha)}_{n}\left(\cos\theta\right)}{P^{(\alpha,\alpha)}_{n}\left(1\right)}=\frac{C^{(\alpha+\frac{1}{2})}_{n}\left(\cos\theta\right)}{C^{(\alpha+\frac{1}{2})}_{n}\left(1\right)}=\frac{2^{\alpha+\frac{1}{2\Gamma\left(\alpha+1\right)}\Gamma\left(\alpha+\frac{1}{2}\right)}(\sin\theta)^{-2\alpha}\int_{0}^{\theta}\frac{\cos\left((n+\alpha+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{-\alpha+\frac{1}{2}\,\mathrm{d}\phi,</math>Laplace-type integral representation<math display="block">\begin{aligned}
\frac{P_n^{(\alpha, \alpha)}(\cos \theta)}{P_n^{(\alpha, \alpha)}(1)} & =\frac{C_n^{\left(\alpha+\frac{1}{2}\right)}(\cos \theta)}{C_n^{\left(\alpha+\frac{1}{2}\right)}(1)} \\
& =\frac{\Gamma(\alpha+1)}{\pi^{\frac{1}{2 \Gamma\left(\alpha+\frac{1}{2}\right)} \int_0^\pi(\cos \theta+i \sin \theta \cos \phi)^n(\sin \phi)^{2 \alpha} \mathrm{~d} \phi
\end{aligned}</math>Addition formula:
<math display="block">\begin{aligned}
& C_n^\lambda\left(\cos \theta_1 \cos \theta_2+\sin \theta_1 \sin \theta_2 \cos \phi\right) \\
& \quad=\sum_{k=0}^n a_{n, k}^\lambda\left(\sin \theta_1\right)^k C_{n-k}^{\lambda+k}\left(\cos \theta_1\right)\left(\sin \theta_2\right)^k C_{n-k}^{\lambda+k}\left(\cos \theta_2\right) \\
& \quad \cdot C_k^{\lambda-1 / 2}(\cos \phi), \quad a_{n, k}^\lambda \text { constants }
\end{aligned}</math>
Asymptotics
Given fixed <math>\lambda \in (0, 1), M \in \{1, 2, \dots\}, \delta \in (0, \pi/2)</math>, uniformly for all <math>\theta\in[\delta,\pi-\delta]</math>, for <math>n \to \infty</math>,<math display="block">C^{(\lambda)}_{n}\left(\cos\theta\right)=
\frac{2^{2\lambda}\Gamma\left(\lambda+\frac{1}{2}\right)}\Gamma\left(\lambda+1\right)}\frac\left(\sum_{m=0}^{M-1}\dfrac{\left(1-\lambda\right)_{m}{m!\,{\left(n+\lambda+1\right)_{m}\dfrac{\cos\theta_{n,m{(2\sin\theta)^{m+\lambda+R_M(\theta)\right)
</math>
where <math>(\cdot)_m</math> is the Pochhammer symbol, and<math display="block">\theta_{n,m}=(n+m+\lambda)\theta-\tfrac{1}{2}(m+\lambda)\pi</math>The remainder <math>R_M = O\left(\frac{1}{n^{M\right)
</math> has an explicit upper bound:<math display="block">|R_M(\theta)| \leq (2 / \pi) \sin (\lambda \pi)
\frac{\Gamma(n+2 \lambda)}{\Gamma(\lambda)} \frac{\Gamma(M+\lambda) \Gamma(M-\lambda+1)}{M!\Gamma(n+M+\lambda+1)} \frac{\max \left(|\cos \theta|^{-1}, 2 \sin \theta\right)}{(2 \sin \theta)^{M+\lambda</math>where <math>\Gamma</math> is the Gamma function.
Other asymptotic formulas can be obtained as special cases of asymptotic formulas for the more general Jacobi polynomials.
See also
- Rogers polynomials, the q-analogue of Gegenbauer polynomials
- Chebyshev polynomials
- Romanovski polynomials