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In mathematics, Meixner polynomials (also called discrete Laguerre polynomials) are a family of discrete orthogonal polynomials introduced by Josef Meixner . They are given in terms of binomial coefficients and the (rising) Pochhammer symbol by
Ruffini's rule can be used when one needs the quotient of a polynomial P by a binomial of the form . (When one needs only the remainder, the polynomial remainder theorem provides a simpler method.) A typical example, where one needs the quotient, is the factorization of a polynomial p ( x ) {\displaystyle p(x)} for which one knows a root r :
This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials (more correctly, irreducible polynomials) can be defined as non-zero polynomials which cannot be factorized into the product of two non-constant 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, a sequence of discrete orthogonal polynomials is a sequence of polynomials that are pairwise orthogonal with respect to a discrete measure. Examples include the discrete Chebyshev polynomials , Charlier polynomials , Krawtchouk polynomials , Meixner polynomials , dual Hahn polynomials , Hahn polynomials , and Racah polynomials .
In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when corresponding coefficients are equal for each different type of term.
Plot of the Jacobi polynomial function (,) with = and = and = in the complex plane from to + with colors created with Mathematica 13.1 function ComplexPlot3D In mathematics , Jacobi polynomials (occasionally called hypergeometric polynomials ) P n ( α , β ) ( x ) {\displaystyle P_{n}^{(\alpha ,\beta )}(x)} are a class of classical orthogonal ...
The identity class in the group is the unique class containing all forms + +, i.e., with first coefficient 1. (It can be shown that all such forms lie in a single class, and the restriction Δ ≡ 0 or 1 ( mod 4 ) {\displaystyle \Delta \equiv 0{\text{ or }}1{\pmod {4}}} implies that there exists such a form of every discriminant.)