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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 ().
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]
When p is a non-negative integer, one or the other of the two functions has its series terminate after a finite number of terms: F terminates if p is even, and G terminates if p is odd. In this case, that function is a polynomial of degree p and it is proportional to the Chebyshev polynomial of the first kind
This proves that for any polynomial h(x) of degree 2n − 1 or less, its integral is given exactly by the Gaussian quadrature sum. To prove the second part of the claim, consider the factored form of the polynomial p n. Any complex conjugate roots will yield a quadratic factor that is either strictly positive or strictly negative over the ...
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.
The Chebyshev functions, especially the second one ψ (x), are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, π (x) (see the exact formula below.) Both Chebyshev functions are asymptotic to x, a statement equivalent to the prime number theorem. Tchebycheff ...
Assuming the equation is defined on the ... It is clear that the eigenvalues of our problem will be the zeros of the nth Chebyshev polynomial of the second kind, ...
Specifically, for α = −1, the Dickson polynomials of the first kind are Fibonacci polynomials, and Dickson polynomials of the second kind are Lucas polynomials. By the composition rule above, when α is idempotent, composition of Dickson polynomials of the first kind is commutative. The Dickson polynomials with parameter α = 0 give monomials.