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Plot of the Chebyshev polynomial of the first kind () with = in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as () and ().
Here we plot the Chebyshev nodes of the first kind and the second kind, both for n = 8. For both kinds of nodes, we first plot the points equi-distant on the upper half unit circle in blue. Then the blue points are projected down to the x-axis. The projected points, in red, are the Chebyshev nodes.
Chebyshev's equation is the second order linear differential equation + = where p is a real (or complex) constant. The equation is named after Russian mathematician Pafnuty Chebyshev. The solutions can be obtained by power series:
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The Dickson polynomials with parameter α = 1 are related to Chebyshev polynomials T n (x) = cos (n arccos x) of the first kind by [1] (,) = (). Since the Dickson polynomial D n (x,α) can be defined over rings with additional idempotents, D n (x,α) is often not related to a Chebyshev polynomial.
For many years he was an associate editor for the Journal of Approximation Theory and wrote over 80 research articles on approximation theory and computational mathematics. [1] The Annals of Numerical Analysis published in 1997 a special issue entitled The Heritage of P.L. Chebyshev: A Festschrift in honor of the 70th birthday of T.J. Rivlin. [4]
In numerical analysis, the Clenshaw algorithm, also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. [1] [2] The method was published by Charles William Clenshaw in 1955. It is a generalization of Horner's method for evaluating a linear combination of monomials.
Somewhat more general Laguerre polynomial sequences are orthogonal with respect to gamma distributions. The Chebyshev polynomials of the first kind are orthogonal with respect to the measure . The Chebyshev polynomials of the second kind are orthogonal with respect to the Wigner semicircle distribution.