Hermite Polynomials

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Hermite polynomials

[er′mēt ‚päl·ə′nō·mē·əlz]
(mathematics)
A family of orthogonal polynomials which arise as solutions to Hermite's differential equation, a particular case of the hypergeometric differential equation.

Hermite Polynomials

 

a special system of polynomials of successively increasing degree. For n = 0,1, 2,..., the Hermite polynomials Hn (x) are defined by the formula

In particular, H0 = 1, H1 = 2x, H2 = 4x2 – 2, H3 = 8x3 – 12x, and H4 = 16x4 – 48x2 + 12.

Hermite polynomials are orthogonal on the entire x-axis with respect to the weight function

e –x2

(seeORTHOGONAL POLYNOMIAL). They satisfy the differential equation

y″ – 2xy′ + 2ny = 0

and the recursion formulas

Hn+1 (x ) – 2xHn (x ) + 2nHn-1 (x ) = 0

H′n(x ) – 2nH–1(x ) = 0

Also sometimes called Hermite polynomials are polynomials that differ from those given above by certain factors dependent on n; sometimes

e–x2/2

is used as the weight function. The basic properties of the system were studied by P. L. Chebyshev in 1859 and C. Hermite in 1864.

References in periodicals archive ?
Half normal was paired with cosine and Hermite polynomial adjustments following Buckland et al.
On specializing the coefficients A(N,k) suitably, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] yields a number of known polynomials such as Laguerre polynomial, Hermite polynomial, Jacobi polynomial, Brafman polynomial, Gould and Hopper polynomials and others.
where x, x' [member of] R and the m-th Hermite polynomial is denoted as [H.
The Hermite polynomial is associated with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
n] denotes the Hermite polynomial of degree n, for which
Barycentric expressions for trigonometric Hermite polynomials are presented in both papers.
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where they give rise to the eigenstates of the quantum harmonic oscillator.
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Hopper, Operational formulas connected with two genaralization of Hermite polynomials, Duke Mathematical journal, 29, pp.
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al 1992) (FLN) has a flat architecture, with predefined basis functions like the trigonometric functions, algebraic polynomials, chebyshev polynomials and Hermite polynomials.
These include among others, the Leguerre polynomials, the Jacobi polynomials, the Hermite polynomials, the Brafman polynomials and several others [16, pp.