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 − "x"2)"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.
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 and harmonic analysis. The Newtonian potential in R"n" 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 formula_14 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