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In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials , consisting of the Hermite polynomials , the Laguerre polynomials and ...
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
The study of Riordan arrays is a field influenced by and contributing to other areas such as combinatorics, group theory, matrix theory, number theory, probability, sequences and series, Lie groups and Lie algebras, orthogonal polynomials, graph theory, networks, unimodal sequences, combinatorial identities, elliptic curves, numerical ...
In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel and Jean Gaston Darboux .
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: signal processing as Hermitian wavelets for wavelet transform analysis; probability, such as the Edgeworth series, as well as in connection with Brownian motion; combinatorics, as an example of an Appell sequence, obeying the umbral ...
Setting the initial values of the sequence equal to this vector produces a geometric sequence = which satisfies the recurrence. In the case of n distinct eigenvalues, an arbitrary solution a k {\displaystyle a_{k}} can be written as a linear combination of such geometric solutions, and the eigenvalues of largest complex norm give an asymptotic ...
Full details on how to obtain expressions for the orthogonal polynomials and the relationship between the coefficients b and a are given by Guest. [2] Expressions for the convolution coefficients are easily obtained because the normal equations matrix, J T J, is a diagonal matrix as the product of any two orthogonal polynomials is zero by ...
Various polynomial sequences named for mathematicians of the past are sequences of orthogonal polynomials. In particular: The Hermite polynomials are orthogonal with respect to the Gaussian distribution with zero mean value. The Legendre polynomials are orthogonal with respect to the uniform distribution on the interval [,].