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
• 
  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 (Stein & Weiss 1971, §IV.2):

${\displaystyle {\frac {1}{(1-2xt+t^{2})^{\alpha }}}=\sum _{n=0}^{\infty }C_{n}^{(\alpha )}(x)t^{n}\qquad (0\leq |x|<1,|t|\leq 1,\alpha >0)}$

• The polynomials satisfy the recurrence relation (Suetin 2001):

${\displaystyle {\begin{aligned}C_{0}^{(\alpha )}(x)&=1\\C_{1}^{(\alpha )}(x)&=2\alpha x\\(n+1)C_{n+1}^{(\alpha )}(x)&=2(n+\alpha )xC_{n}^{(\alpha )}(x)-(n+2\alpha -1)C_{n-1}^{(\alpha )}(x).\end{aligned}}}$

• Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation (Suetin 2001):

${\displaystyle (1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0.\,}$

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.^[1]

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

${\displaystyle C_{n}^{(\alpha )}(z)={\frac {(2\alpha )_{n}}{n!}}\,_{2}F_{1}\left(-n,2\alpha +n;\alpha +{\frac {1}{2}};{\frac {1-z}{2}}\right).}$

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

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

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

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

• They are special cases of the Jacobi polynomials (Suetin 2001):

${\displaystyle C_{n}^{(\alpha )}(x)={\frac {(2\alpha )_{n}}{(\alpha +{\frac {1}{2}})_{n}}}P_{n}^{(\alpha -1/2,\alpha -1/2)}(x).}$

in which ${\displaystyle (\theta )_{n}}$ represents the rising factorial of ${\displaystyle \theta }$.

One therefore also has the Rodrigues formula

${\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-1)^{n}}{2^{n}n!}}{\frac {\Gamma (\alpha +{\frac {1}{2}})\Gamma (n+2\alpha )}{\Gamma (2\alpha )\Gamma (\alpha +n+{\frac {1}{2}})}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right].}$

• An alternative normalization sets ${\displaystyle C_{n}^{(\alpha )}(1)=1}$. Assuming this alternative normalization, the derivatives of Gegenbauer are expressed in terms of Gegenbauer:^[2]

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

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)

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

To wit, for n ≠ m,

${\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx=0.}$

They are normalized by

${\displaystyle \int _{-1}^{1}\left[C_{n}^{(\alpha )}(x)\right]^{2}(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}.}$

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,

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

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 (Stein & Weiss 1971).

It follows that the quantities ${\displaystyle C_{k}^{((n-2)/2)}(\mathbf {x} \cdot \mathbf {y} )}$ 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

${\displaystyle \sum _{j=0}^{n}{\frac {C_{j}^{\alpha }(x)}{2\alpha +j-1 \choose j}}\geq 0\qquad (x\geq -1,\,\alpha \geq 1/4).}$

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.^[3]

Other properties

Dirichlet–Mehler-type integral representation:^[4]$${\displaystyle {\frac {P_{n}^{(\alpha ,\alpha )}\left(\cos \theta \right)}{P_{n}^{(\alpha ,\alpha )}\left(1\right)}}={\frac {C_{n}^{(\alpha +{\frac {1}{2}})}\left(\cos \theta \right)}{C_{n}^{(\alpha +{\frac {1}{2}})}\left(1\right)}}={\frac {2^{\alpha +{\frac {1}{2}}}\Gamma \left(\alpha +1\right)}{{\pi }^{\frac {1}{2}}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}(\sin \theta )^{-2\alpha }\int _{0}^{\theta }{\frac {\cos \left((n+\alpha +{\tfrac {1}{2}})\phi \right)}{(\cos \phi -\cos \theta )^{-\alpha +{\frac {1}{2}}}}}\,\mathrm {d} \phi ,}$$Laplace-type$${\displaystyle {\begin{aligned}{\frac {P_{n}^{(\alpha ,\alpha )}(\cos \theta )}{P_{n}^{(\alpha ,\alpha )}(1)}}&={\frac {C_{n}^{\left(\alpha +{\frac {1}{2}}\right)}(\cos \theta )}{C_{n}^{\left(\alpha +{\frac {1}{2}}\right)}(1)}}\\&={\frac {\Gamma (\alpha +1)}{\pi ^{\frac {1}{2}}\Gamma \left(\alpha +{\frac {1}{2}}\right)}}\int _{0}^{\pi }(\cos \theta +i\sin \theta \cos \phi )^{n}(\sin \phi )^{2\alpha }\mathrm {~d} \phi \end{aligned}}}$$

See also

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