Search results
Results From The WOW.Com Content Network
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.
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.
In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain , the bilinear form may be the integral of the product of functions over the interval:
Suppose that y 0 = 1, y 1, ... is a sequence of polynomials where y n has degree n. If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a 3-term recurrence relation. Favard's theorem is roughly a converse of this, and states that if these polynomials satisfy a 3-term recurrence relation of the form
Let (()) = be a sequence of orthogonal polynomials defined on the interval [,] satisfying the orthogonality condition () =,, where () is a suitable weight function, is a constant depending on , and , is the Kronecker delta.
Askey, Richard; Ismail, Mourad E. H.; Van Assche, Walter (2001), "Ted Chihara and his work on orthogonal polynomials", Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), Journal of Computational and Applied Mathematics, 133 (1): 1– 11, doi: 10.1016/S0377-0427(00)00631-2, ISSN 0377-0427, MR 1858266
The summation involved in the Christoffel–Darboux formula is invariant by scaling the polynomials with nonzero constants. Thus, each probability distribution defines a series of functions (,):= = () /, =,, … which are called the Christoffel–Darboux kernels.
In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilson that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by