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.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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

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References in periodicals archive

Now, we define the generalized Jacobi polynomials of degree n with a, ff G Z on interval [c, d] by

Generalized Jacobi Reproducing Kernel Method in Hilbert Spaces for Solving the Black-Scholes Option Pricing Problem Arising in Financial Modelling

[7] , On the zeros of subrange Jacobi polynomials, Numer.

BINET-TYPE POLYNOMIALS AND THEIR ZEROS

Association of Rodrigues Representation with Jacobi Polynomials

Spinors and Rodrigues Representations Associated with Orthogonal Polynomials

In the theory of classical orthogonal polynomials, the asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials are well known.

A Transference Result of the [L.sup.p]-Continuity of the Jacobi Littlewood-Paley g-Function to the Gaussian and Laguerre Littlewood-Paley g-Function

More recently, the authors [38] developed some new results related to the Jacobi polynomials for operational matrices.

Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

J-GL-C methods are based on Jacobi polynomials. The Jacobi polynomials are the eigenfunctions of the singular Sturm-Liouville problem [8],

Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs

As we know, classical Jacobi polynomials [48] are defined on the interval of [phi] [member of] [-1, 1] and the recurrence formula of the classical Jacobi polynomials [P.sup.([alpha],[beta]).sub.i]([phi]) of degree i is given by

Free and Forced Vibration Analysis of Airtight Cylindrical Vessels with Doubly Curved Shells of Revolution by Using Jacobi-Ritz Method

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.

The Classical Orthogonal Polynomials

Pavlovic, "New class of filter functions generated directly by the modified Christoffel-Darboux formula for classical orthonormal Jacobi polynomials", Int.

Transitional selective linear phase 1D FIR filter function generated by Christoffel-Darboux formula for Chebyshev polynomials

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.

A numerical approach for the solution of nonlinear boundary value problems arising in biology via shifted Jacobi operational matrix