Gegenbauer polynomials


Also found in: Wikipedia.

Gegenbauer polynomials

[′gāg·ən‚bau̇r ‚päl·i′nō·mē·əlz]
(mathematics)
A family of polynomials solving a special case of the Gauss hypergeometric equation. Also known as ultraspherical polynomials.
References in periodicals archive ?
One of the formulas is given in [5, Theorem 1.1] and expressed as an infinite sum of which each summand consists of the modified Bessel functions, the Gegenbauer polynomials and the densities [p.sub.r.sup(0)] (.;x).
The Jacoby polynomials are generalized orthonormal polynomials containing some orthonormal polynomials such as Legendre, Chebyshev, and Gegenbauer polynomials. For example, the choice [alpha] = [beta] = -1/2 yields the Chebyshev polynomials of the first kind, while choosing [alpha] = [beta] = 1/2 gives the Chebyshev polynomials of the second kind.
Author Brian George Spencer Doman examines classical orthogonal polynomials and their additional properties, covering hermite polynomials, associated Laguerre polynomials, Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials, associated Legendre functions, Jacobi polynomials, and many other related mathematical subjects over twelve chapters and appendices.
Gegenbauer polynomials or ultraspherical polynomials [C.sup.[alpha].sub.n](x) are orthogonal polynomials on the interval [-1,1] with respect to the weight function [(1-[x.sup.2]).sup.[alpha]-1/2].
This can be done for the unispherical windows [3] based on Gegenbauer polynomials as well as for windows proposed by Zierhofer [4].
For a given positive integer number N, the orthogonality of the generalized Gegenbauer polynomials [{[C.sup.(-N+1/2).sub.n]}.sup.[infinity].sub.n=0] is solved in [3], using again a Sobolev inner product, that is, an inner product involving derivatives (the case N = 1 is considered also in [12]).
The Gegenbauer polynomials of degree m denoted by [C.sup.[LAMBDA].sub.m] differ from each other through a parameter [LAMBDA].
Stieltjes interlacing was studied for the zeros of polynomials from different sequences of one-parameter orthogonal families, namely, Gegenbauer polynomials [C.sup.[lambda].sub.n] and Laguerre polynomials [L.sup.[alpha].sub.n] in [5] and [6], respectively, and associated polynomials analogous to the de Boor-Saff polynomials were identified in each case.
Key words and phrases : Gibbs phenomenon, Gegenbauer polynomials, spherical harmonics, edge detection.
Generalizations of the classical Gegenbauer polynomials to the Clifford analysis framework are called Clifford-Gegenbauer polynomials and were introduced as well on the closed unit ball B(1) (see [4]), as on the Euclidean space [R.sup.m] (see [2,8]).
This also holds for Gegenbauer polynomials where sines and cosines (in this order) are included in the form of Chebyshev polynomials of second and first kind.
which plays an important role in the study of orthogonal expansions in Gegenbauer polynomials; see, for example, [4, 5, 7, 18, 22].