Hermite Polynomials

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Hermite polynomials

[er′mēt ‚päl·ə′nō·mē·əlz]
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


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 ?
1, the following representation holds for the Hermite polynomials (1.
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.
De Bie, An alternative definition of the Hermite polynomials related to the Dunkl laplacian.
Rezola, Asymptotics for a generalization of Hermite polynomials, Asymp.
An appendix provides background on hermite polynomials and a reference on fundamental constants.
Hopper, Operational formulas connected with two genaralization of Hermite polynomials, Duke Mathematical journal, 29, pp.
As Fujiwara [30] showed, the most important property of the Jacobi, Laguerre, and Hermite polynomials is the generalized Rodriguez formula.
Some generating functions for Laguerre and Hermite polynomials, Canad.
al 1992) (FLN) has a flat architecture, with predefined basis functions like the trigonometric functions, algebraic polynomials, chebyshev polynomials and Hermite polynomials.
11) are seen to obey a recursion relation that is equivalent to the standard one between even order Hermite polynomials.
For the univariate case, Gallant and Nychka [Econometrica, 1987] proposed a straightforward solution based on squaring the weighted sum of Hermite polynomials.