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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.svg|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 File:Mplwp gegenbauer Cn05a1.svg|Gegenbauer polynomials with ''α''=1 File:Mplwp gegenbauer Cn05a2.svg|Gegenbauer polynomials with ''α''=2 File:Mplwp gegenbauer Cn05a3.svg|Gegenbauer polynomials with ''α''=3 File:Gegenbauer polynomials.gif|An animation showing the polynomials on the ''xα''-plane for the first 4 values of ''n''. A variety of characterizations of the Gegenbauer polynomials are available. The polynomials can be defined in terms of their generating function : The polynomials satisfy the recurrence relation : Gegenbauer polynomials are particular solutions of the Gegenbauer differential equation : When α = 1/2, the equation reduces to the Legendre equation, and the Gegenbauer polynomials reduce to the Legendre polynomials. When α = 1, the equation reduces to the Chebyshev differential equation, and the Gegenbauer polynomials reduce to the Chebyshev polynomials of the second kind. They are given as Gaussian hypergeometric series in certain cases where the series is in fact finite: (Abramowitz & Stegun p. 561). Here (2α)n is the rising factorial. Explicitly, From this it is also easy to obtain the value at unit argument: They are special cases of the Jacobi polynomials : in which represents the rising factorial of . One therefore also has the Rodrigues formula For a fixed α > -1/2, the polynomials are orthogonal on [−1, 1] with respect to the weighting function (Abramowitz & Stegun p.

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Jacobi polynomials

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).

Chebyshev polynomials

The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind are defined by Similarly, the Chebyshev polynomials of the second kind are defined by That these expressions define polynomials in may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly.

Orthogonal polynomials

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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