Gegenbauer polynomials

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

• 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:

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

• 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:

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

Here (2α)_n is the rising factorial. Explicitly,

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

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

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

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

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

Orthogonality and normalization

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

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

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

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

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,

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

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 $C_k^{((n-2)/2)}(𝐱\cdot 𝐲)$ 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:$$\frac{P_n^{(\alpha ,\alpha )}(\cos \theta )}{P_n^{(\alpha ,\alpha )}\left(1\right)}=\frac{C_n^{(\alpha +\frac{1}{2})}(\cos \theta )}{C_n^{(\alpha +\frac{1}{2})}\left(1\right)}=\frac{2^{\alpha +\frac{1}{2}}\Gamma (\alpha +1)}{{\pi }^{\frac{1}{2}}\Gamma (\alpha +\frac{1}{2})}(\sin \theta {)}^{-2\alpha }\underset{0}{\overset{\theta }{\int }}\frac{\cos ((n+\alpha +\frac{1}{2})\varphi )}{(\cos \varphi -\cos \theta {)}^{-\alpha +\frac{1}{2}}}\mathrm{d}\varphi ,$$Laplace-type integral representation$$\begin{array}{cc}\frac{P_n^{(\alpha ,\alpha )}(\cos \theta )}{P_n^{(\alpha ,\alpha )}(1)}& =\frac{C_n^{(\alpha +\frac{1}{2})}(\cos \theta )}{C_n^{(\alpha +\frac{1}{2})}(1)}\\ & =\frac{\Gamma (\alpha +1)}{{\pi }^{\frac{1}{2}}\Gamma (\alpha +\frac{1}{2})}\underset{0}{\overset{\pi }{\int }}(\cos \theta +i\sin \theta \cos \varphi {)}^n(\sin \varphi {)}^{2\alpha }\mathrm{d}\varphi \end{array}$$Addition formula:

$$\begin{array}{cc}& C_n^{\lambda }(\cos {\theta }_1\cos {\theta }_2+\sin {\theta }_1\sin {\theta }_2\cos \varphi )\\ & =\sum _{k=0}^n{a}_{n,k}^{\lambda }{(\sin {\theta }_1)}^k{C}_{n-k}^{\lambda +k}(\cos {\theta }_1){(\sin {\theta }_2)}^k{C}_{n-k}^{\lambda +k}(\cos {\theta }_2)\\ & \cdot C_k^{\lambda -1/2}(\cos \varphi ),a_{n,k}^{\lambda }\text{ constants }\end{array}$$

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 $n\to \mathrm{\infty }$,$$C_n^{(\lambda )}(\cos \theta )=\frac{2^{2\lambda }\Gamma (\lambda +\frac{1}{2})}{{\pi }^{\frac{1}{2}}\Gamma (\lambda +1)}\frac{{\left(2\lambda \right)}_n}{{(\lambda +1)}_n}(\sum _{m=0}^{M-1}\frac{{\left(\lambda \right)}_m{(1-\lambda )}_m}{m!{(n+\lambda +1)}_m}\frac{\cos {\theta }_{n,m}}{(2\sin \theta {)}^{m+\lambda }}+R_M(\theta ))$$

where $(\cdot {)}_m$ is the Pochhammer symbol, and$${\theta }_{n,m}=(n+m+\lambda )\theta -\frac{1}{2}(m+\lambda )\pi$$The remainder $R_M=O\left(\frac{1}{n^M}\right)$ has an explicit upper bound:$$|R_M(\theta )|\le (2/\pi )\sin (\lambda \pi )\frac{\Gamma (n+2\lambda )}{\Gamma (\lambda )}\frac{\Gamma (M+\lambda )\Gamma (M-\lambda +1)}{M!\Gamma (n+M+\lambda +1)}\frac{\max (|\cos \theta {|}^{-1},2\sin \theta )}{(2\sin \theta {)}^{M+\lambda }}$$where $\Gamma$ is the Gamma function.

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

See also

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

References

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