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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
Moreover, they proved that the spaces [F.sup.2.sub.(n)](C) are isomorphic and isometric to [L.sub.2](R) [R] [cross product] [H.sub.n-1], where [H.sub.n-1] is the one-dimensional space generated by Hermite function of order n - 1.
Parand, Using Hermite Function for Solving Thomas-Fermi Equation, Int.
In this determination, we do not care about the optimum initialization of the POPS algorithm and stick to the Gaussian function, as an initialization of the transmit waveform, which is also the first Hermite function, that offers the best localization on the TF plane.
Parand, "Using hermite function for solving thomas-fermi equation," International Journal of Mathematical, Computational Science and Engineering, vol.
Despite slight differences in the two most parsimonious models in the 5%truncation dataset, these models are essentially the same; the hermite function uses an additional parameter to fit the detection function, and this additional parameter penalizes the AIC value by 2 ([DELTA]AIC = 2, Table 1).
The asymptotic coefficients of Hermite function series.
The DFRFT matrix is generated through calculating the power of the discrete Fourier matrix, which involves the discrete Hermite function. However, the discrete Hermite function cannot be expressed in analytical form.
the corresponding Hermite function. Then the set {[e.sub.n], n [greater than or equal to] 0} is an orthonormal basis for the Hilbert space [L.sup.2]([??]).
Wang and Jiao [27] considered the Fisher equation on unbounded domain using the generalized Hermite functions.
The ordinary hypergeometric [19-25], confluent hypergeometric [26-29], Coulomb wave functions [30,31], Bessel and Hankel functions [32], incomplete Beta and Gamma functions [33-35], Hermite functions [36,37], Goursat and Appell generalized hypergeometric functions of two variables of the first kind [38,39], and other known special functions have been used as expansion functions.
The Hermite functions, which were widely studied in the Hermite spectral method (HSM), are constructed by Hermite basis functions with a translated and scaled weighting function [35].