Gegenbauer polynomials

Template:Use American English Template:Short description In mathematics, Gegenbauer polynomials or ultraspherical polynomials CTemplate:Su(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
• 
  Gegenbauer polynomials with α=1
• 
  Gegenbauer polynomials with α=2
• 
  Gegenbauer polynomials with α=3
• 
  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 Template:Harv:

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

• The polynomials satisfy the recurrence relation Template:Harv:

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

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

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

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 Template:Harv:

$C_n^{(\alpha )}(x)=\frac{(2\alpha {)}_n}{(\alpha +\frac{1}{2}{)}_n}{P}_n^{(\alpha -1/2,\alpha -1/2)}(x).$

in which $(\theta {)}_n$ represents the rising factorial of $\theta$.

One therefore also has the Rodrigues formula

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

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,

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

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 Template:Harv.

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.

See also

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

References

• Template:Abramowitz Stegun ref*Template:Dlmf
• Template:Citation.
• Template:Springer.

Specific