Jacobi Polynomials


Also found in: Wikipedia.

Jacobi polynomials

[jə′kō·bē ‚päl·ə′nō·mē·əlz]
(mathematics)
Polynomials that are constructed from the hypergeometric function and satisfy the differential equation (1 - x 2) y ″ + [β - α - (α + β + 2) x ] y ′ + n (α + β + n + 1) y = 0, where n is an integer and α and β are constants greater than -1; in certain cases these generate the Legendre and Chebyshev polynomials.

Jacobi Polynomials

 

a special system of polynomials of successively increasing degree. For n = 0, 1, 2, . . . the Jacobi polynomials Jacobi Polynomials (x) may be defined by the formula

Jacobi polynomials are orthogonal on the interval [–1, 1] with respect to the weight function (1 – x)α(l + x)β (seeORTHOGONAL POLYNOMIAL). They were introduced by K. Jacobi in a work published in 1859. Legendre polynomials (α = β = 0), Chebyshev polynomials of the first kind (α = β = –½) and of the second kind (α = β = ½), and ultraspherical polynomials (α = β) are special cases of Jacobi polynomials. Jacobi polynomials are a particular case of the hypergeometric function. They satisfy the differential equation

(1 + x2)y +[β – α – (α + β + 2)x]y +n(α + β +n + 1)y = 0

References in periodicals archive ?
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.
7] proposed a computational method based on Jacobi Polynomials for solving fuzzy linear FDE on interval [0,1].
I, 1826) extended it to Jacobi polynomials twelve years later.
In this paper by use of shifted Jacobi polynomials as basis and operational matrix of derivatives [23], of them we convert these kinds of equations to nonlinear algebraic equations.
Also, in [12]-[15], we proposed a form of a prototype class of low frequency selective polynomial analogue filter functions, derived in compact explicit form on a simple way by direct application of the modified Christoffel-Darboux formula for classical continual orthonormal Jacobi polynomials [16], [17].
They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials.
Other cases for Jacobi polynomials are solved in [2], where the families [{[[P.
There, the authors obtained a connection formula that relates the Jacobi Sobolev--type polynomials with some family of Jacobi polynomials, as well as the holonomic equation that they satisfy.
One example of operator is the Rodriguez operator operating on holomorphic functions, this is a generalization of the Rodriguez formula for Laguerre and Jacobi polynomials.
These include among others, the Leguerre polynomials, the Jacobi polynomials, the Hermite polynomials, the Brafman polynomials and several others [16, pp.
It is well known that the spectral Tau method based on the classical Jacobi polynomials (Jacobi Tau method) allows the approximation of infinitely smooth solutions of operator equations such that the truncation error approaches zero faster than any negative power of the number of basis functions used in the approximation as that number tends to [infinity].
Guldan, "On Jacobi Polynomials and Related Functions, in Proc.