Jacobi Polynomials

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 - x²)y″ + [β - α - (α + β + 2)x]y′ +n(α + β +n+ 1)y= 0, wherenis an integer and α and β are constants greater than -1; in certain cases these generate the Legendre and Chebyshev polynomials.

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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