Gegenbauer polynomials

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

• They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite:

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:

:in which (\theta)_n represents the rising factorial of \theta. :One therefore also has the Rodrigues formula

• An alternative normalization sets C_n^{(\alpha)}(1)=1. Assuming this alternative normalization, the derivatives of Gegenbauer are expressed in terms of Gegenbauer: \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}

Orthogonality and normalization

For a fixed α -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function

: 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 − 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.

Other properties

Dirichlet–Mehler-type integral representation:\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,Laplace-type integral representation\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}Addition formula:

\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}

Asymptotics

Given fixed \lambda \in (0, 1), M \in {1, 2, \dots}, \delta \in (0, \pi/2), uniformly for all \theta\in[\delta,\pi-\delta], for n \to \infty,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)

where (\cdot)m is the Pochhammer symbol, and\theta{n,m}=(n+m+\lambda)\theta-\tfrac{1}{2}(m+\lambda)\piThe remainder R_M = O\left(\frac{1}{n^{M}}\right) has an explicit upper bound:|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}}where \Gamma is the Gamma function.

Other asymptotic formulas can be obtained as special cases of asymptotic formulas for the more general Jacobi polynomials.

References

Specific

References

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5. Doha, E. H.. (1991-01-01). "The coefficients of differentiated expansions and derivatives of ultraspherical polynomials". Computers & Mathematics with Applications.
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