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In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product | = () ¯ where dμ is a given positive measure on [−π, π].
The Macdonald polynomials are orthogonal polynomials in several variables, depending on the choice of an affine root system. They include many other families of multivariable orthogonal polynomials as special cases, including the Jack polynomials, the Hall–Littlewood polynomials, the Heckman–Opdam polynomials, and the Koornwinder polynomials.
Gábor Szegő (Hungarian: [ˈɡaːbor ˈsɛɡøː]) (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician.He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and Toeplitz matrices building on the work of his contemporary Otto Toeplitz.
In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle in the complex plane, for some probability measure on the unit circle. They were introduced by Szegő (1920, 1921, 1939).
In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Szegő , who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers. They are given by
Chihara, Theodore Seio (2001), "45 years of orthogonal polynomials: a view from the wings", Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), Journal of Computational and Applied Mathematics, 133 (1): 13– 21, doi: 10.1016/S0377-0427(00)00632-4, ISSN 0377-0427 ...
Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.
Note that this will be true for all the orthogonal polynomials above, because each p n is constructed to be orthogonal to the other polynomials p j for j<n, and x k is in the span of that set. If we pick the n nodes x i to be the zeros of p n , then there exist n weights w i which make the Gaussian quadrature computed integral exact for all ...