Gegenbauer polynomials

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

A variety of characterizations of the Gegenbauer polynomials are available.

• The polynomials can be defined in terms of their generating function:

:

• The polynomials satisfy the recurrence relation:

:

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

:

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:

:

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

:

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

:

• They are special cases of the Jacobi polynomials:

:

in which represents the rising factorial of .

One therefore also has the Rodrigues formula

:

• An alternative normalization sets . Assuming this alternative normalization, the derivatives of Gegenbauer are expressed in terms of Gegenbauer:

Orthogonality and normalization

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

To wit, for n Ã¢Â‰Â  m,

They are normalized by

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,

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 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

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:Laplace-type integral representationAddition formula:

Asymptotics

Given fixed , uniformly for all , for ,

where is the Pochhammer symbol, andThe remainder has an explicit upper bound:where 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

Specific