Gegenbauer polynomials

Template:Use American English Template:Short description In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(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

• 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
  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
• Gegenbauer polynomials with α=1
  Gegenbauer polynomials with α=1
• Gegenbauer polynomials with α=2
  Gegenbauer polynomials with α=2
• Gegenbauer polynomials with α=3
  Gegenbauer polynomials with α=3
• An animation showing the polynomials on the xα-plane for the first 4 values of n.
  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 :

$\frac{1}{(1-2xt+t^2{)}^{\alpha }}=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\alpha )}(x)t^n(0\le |x|<1,|t|\le 1,\alpha >0)$

• The polynomials satisfy the recurrence relation :

$\begin{array}{rl}{C}_0^{(\alpha )}(x)& =1\\ C_1^{(\alpha )}(x)& =2\alpha x\\ (n+1)C_{n+1}^{(\alpha )}(x)& =2(n+\alpha )x{C}_n^{(\alpha )}(x)-(n+2\alpha -1)C_{n-1}^{(\alpha )}(x).\end{array}$

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

$(1-x^2)y^{″}-(2\alpha +1)x{y}^{\prime }+n(n+2\alpha )y=0.$

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:

$C_n^{(\alpha )}(z)=\frac{(2\alpha {)}_n}{n!}{}_2{F}_1(-n,2\alpha +n;\alpha +\frac{1}{2};\frac{1-z}{2}).$

(Abramowitz & Stegun p. 561). Here (2α)_n is the rising factorial. Explicitly,

$C_n^{(\alpha )}(z)=\sum _{k=0}^{\lfloor n/2\rfloor }(-1{)}^k\frac{\mathrm{\Gamma }(n-k+\alpha )}{\mathrm{\Gamma }(\alpha )k!(n-2k)!}(2z{)}^{n-2k}.$

From this it is also easy to obtain the value at unit argument:

$C_n^{(\alpha )}(1)=\frac{\mathrm{\Gamma }(2\alpha +n)}{\mathrm{\Gamma }(2\alpha )n!}.$

• 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

$C_n^{(\alpha )}(x)=\frac{(-1{)}^n}{2^{n}n!}\frac{\mathrm{\Gamma }(\alpha +\frac{1}{2})\mathrm{\Gamma }(n+2\alpha )}{\mathrm{\Gamma }(2\alpha )\mathrm{\Gamma }(\alpha +n+\frac{1}{2})}(1-x^2{)}^{-\alpha +1/2}\frac{d^n}{d{x}^n}[(1-x^2{)}^{n+\alpha -1/2}].$

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

$$\begin{array}{rl}\frac{d^q}{d{x}^q}{C}_{q+2j+1}^{(\alpha )}(x)=\frac{2^q(q+2j+1)!}{(q-1)!\mathrm{\Gamma }(q+2j+2\alpha +1)}& \sum _{i=0}^j\frac{(2i+\alpha +1)\mathrm{\Gamma }(2i+2\alpha +1)}{(2i+1)!(j-i)!}\\ & \times \frac{\mathrm{\Gamma }(q+j+i+\alpha +1)}{\mathrm{\Gamma }(j+i+\alpha +2)}(q+j-i-1)!C_{2i+1}^{(\alpha )}(x)\end{array}$$

Orthogonality and normalization

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

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

To wit, for n ≠ m,

$\underset{-1}{\overset{1}{\int }}{C}_n^{(\alpha )}(x)C_m^{(\alpha )}(x)(1-x^2{)}^{\alpha -\frac{1}{2}}dx=0.$

They are normalized by

$\underset{-1}{\overset{1}{\int }}{[C_n^{(\alpha )}(x)]}^2(1-x^2{)}^{\alpha -\frac{1}{2}}dx=\frac{\pi 2^{1-2\alpha }\mathrm{\Gamma }(n+2\alpha )}{n!(n+\alpha )[\mathrm{\Gamma }(\alpha ){]}^2}.$

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,

$\frac{1}{|\mathbf{𝐱}-\mathbf{𝐲}{|}^{n-2}}=\sum _{k=0}^{\mathrm{\infty }}\frac{|\mathbf{𝐱}{|}^k}{|\mathbf{𝐲}{|}^{k+n-2}}{C}_k^{(\alpha )}(\frac{\mathbf{𝐱}\cdot \mathbf{𝐲}}{|\mathbf{𝐱}||\mathbf{𝐲}|}).$

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 $C_k^{((n-2)/2)}(\mathbf{𝐱}\cdot \mathbf{𝐲})$ 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

$\sum _{j=0}^n\frac{C_j^{\alpha }(x)}{(\genfrac{}{}{0ex}{}{2\alpha +j-1}{j})}\ge 0(x\ge -1,\alpha \ge 1/4).$

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:$$\frac{P_n^{(\alpha ,\alpha )}(\cos \theta )}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}(\cos \theta )}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{2^{\alpha +\frac{1}{2}}\mathrm{\Gamma }(\alpha +1)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}(\sin \theta {)}^{-2\alpha }\underset{0}{\overset{\theta }{\int }}\frac{\cos ((n+\alpha +\frac{1}{2})\varphi )}{(\cos \varphi -\cos \theta {)}^{-\alpha +\frac{1}{2}}}\mathrm{d}\varphi ,$$Laplace-type$$\begin{array}{rl}\frac{P_n^{(\alpha ,\alpha )}(\cos \theta )}{P_n^{(\alpha ,\alpha )}(1)}& =\frac{C_n^{(\alpha +\frac{1}{2})}(\cos \theta )}{C_n^{(\alpha +\frac{1}{2})}(1)}\\ & =\frac{\mathrm{\Gamma }(\alpha +1)}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }(\alpha +\frac{1}{2})}\underset{0}{\overset{\pi }{\int }}(\cos \theta +i\sin \theta \cos \varphi {)}^n(\sin \varphi {)}^{2\alpha }\mathrm{d}\varphi \end{array}$$

See also

• Rogers polynomials, the q-analogue of Gegenbauer polynomials
• Chebyshev polynomials
• Romanovski polynomials

References

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• Template:Springer.

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