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The Chebyshev polynomials of the second kind are defined by the recurrence relation: = = + = (). Notice that the two sets of recurrence relations are identical, except for () = vs. () =.
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:
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.
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
The Chebyshev nodes of the second kind, also called the Chebyshev extrema, are the extrema of the Chebyshev polynomials of the first kind, which are also the zeros of the Chebyshev polynomials of the second kind. Both of these sets of numbers are commonly referred to as Chebyshev nodes in literature. [1] Polynomial interpolants constructed from ...
The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by P. L. Chebyshev and was pursued by A. A. Markov and T. J. Stieltjes.
U n (x, −1) : Fibonacci polynomials V n (x, −1) : Lucas polynomials U n (2x, 1) : Chebyshev polynomials of second kind V n (2x, 1) : Chebyshev polynomials of first kind multiplied by 2 U n (x+1, x) : Repunits in base x V n (x+1, x) : x n + 1. Some Lucas sequences have entries in the On-Line Encyclopedia of Integer Sequences:
A typical example of a Chebyshev space is the subspace of Chebyshev polynomials of order n in the space of real continuous functions on an interval, C[a, b]. The polynomial of best approximation within a given subspace is defined to be the one that minimizes the maximum absolute difference between the polynomial