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Concept# Jacobi polynomials

Summary

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials)
are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight
on the interval . The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.
The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.
The Jacobi polynomials are defined via the hypergeometric function as follows:
where is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:
An equivalent definition is given by Rodrigues' formula:
If , then it reduces to the Legendre polynomials:
For real the Jacobi polynomial can alternatively be written as
and for integer
where is the gamma function.
In the special case that the four quantities , , ,
are nonnegative integers, the Jacobi polynomial can be written as
The sum extends over all integer values of for which the arguments of the factorials are nonnegative.
The Jacobi polynomials satisfy the orthogonality condition
As defined, they do not have unit norm with respect to the weight. This can be corrected by dividing by the square root of the right hand side of the equation above, when .
Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity:
The polynomials have the symmetry relation
thus the other terminal value is
The th derivative of the explicit expression leads to
The Jacobi polynomial is a solution of the second order linear homogeneous differential equation
The recurrence relation for the Jacobi polynomials of fixed , is:
for .
Writing for brevity , and , this becomes in terms of
Since the Jacobi polynomials can be described in terms of the hypergeometric function, recurrences of the hypergeometric function give equivalent recurrences of the Jacobi polynomials.

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In mathematics, Gegenbauer polynomials or ultraspherical polynomials C(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer. File:Plot of the Gegenbauer polynomial C n^(m)(x) with n=10 and m=1 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight on the interval . The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials. The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi. The Jacobi polynomials are defined via the hypergeometric function as follows: where is Pochhammer's symbol (for the rising factorial).

In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases.

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