# Jacobi Polynomials

(redirected from Jacobi polynomial)

## 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 (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 ?
The special case of [beta] = [alpha] of the Jacobi polynomial is called Ultraspherical polynomial and is denoted by [P.
On specializing the coefficients A(N,k) suitably, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] yields a number of known polynomials such as Laguerre polynomial, Hermite polynomial, Jacobi polynomial, Brafman polynomial, Gould and Hopper polynomials and others.
n], [alpha],[beta] > -1, n [member of] N, denote the Jacobi polynomial of degree n.
2) and writing the result in terms of x = cos [theta] and the orthonormal Jacobi polynomial [[?
n] denotes the Jacobi polynomial of degree n with the normalization (4.
alpha],[beta]) denotes the orthonormal Jacobi polynomial of degree n, it follows from (5.
In all cases we show that the orthogonality conditions characterize the Jacobi polynomial [P.
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].

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