Jacobi Polynomials

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Jacobi polynomials

[jə′kō·bē ‚päl·ə′nō·mē·əlz]
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 ?
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.
They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials.
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].
Schur functions can be replaced with characters of symplectic group or, more, generally, with multivariate Jacobi polynomials.
It is clear that our methods can also be used to deal with the other orthogonal polynomials, such as the Legendre polynomials, the Chebyshev polynomials, Jacobi polynomials and the Hermite polynomials, etc.
Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
It is worth quoting that an alternative method for the computation of the recurrence coefficients could resort to the more general sub-range Jacobi polynomials approach studied in [11].
In this paper, we investigate the extent to which Stieltjes interlacing holds between the zeros of two Jacobi polynomials if each polynomial belongs to a sequence generated by a different value of the parameter [alpha] and/or [beta].
Also in each figure the ridge and the one-sided Jacobi polynomials show a faster convergence than the shifted Chebyshev polynomials.