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.

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Here, the pth order Hermite basis function is defined using the pth order Hermite polynomial as follows:
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0] is the correlation length of the source, and H(*) is Hermite polynomial.
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n] denotes the Hermite polynomial of degree n, for which
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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.