# Jacobi Polynomials

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## 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} (*see*ORTHOGONAL 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 + *x*^{2})*y*^{″} +[β – α – (α + β + 2)*x*]*y*^{′} +*n*(α + β +*n* + 1)*y* = 0