Characterizations

• The polynomials can be defined in terms of their generating function
  Generating function
  In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
  
   :

• The polynomials satisfy the recurrence relation
  Recurrence relation
  In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
  
   :

• 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.

• 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,

• They are special cases of the Jacobi polynomials
  Jacobi polynomials
  In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...
  
   :

One therefore also has the Rodrigues formula

Orthogonality and normalization

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


To wit, for n ≠ m,


They are normalized by

Applications

The Gegenbauer polynomials appear naturally as extensions of Legendre polynomials in the context of potential theory

Potential theory

In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

 and harmonic analysis

Harmonic analysis

Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

. The Newtonian potential

Newtonian potential

In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory...

 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

Poisson kernel

In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc. The kernel can be understood as the derivative of the Green's function for the Laplace equation...

 in a ball .

It follows that the quantities are spherical harmonics

Spherical harmonics

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

, 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.

See also

• Rogers polynomials
  Rogers polynomials
  In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by in the course of his work on the Rogers–Ramanujan identities...
  
  , the q-analogue of Gegenbauer polynomials
• Chebyshev polynomials
  Chebyshev polynomials
  In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. One usually distinguishes between Chebyshev polynomials of the first kind which are denoted Tn and...
  
  

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