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 ?
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 most important special functions are known as: Bessel functions, Hermite functions, Legendre functions, Laguerre functions, Chebyshev functions etc.
Subjects include fractional integration and fractional differentiation for d- dimensional Jacobi expansions, Sutherland-type trigonometric models, a generating function for N-soliton solutions of the Kadomtsev- Petviashvili II equation, asymptotics of the second Painleve equation, evaluation of certain Mellin transformations in terms of the trigamma and polygamma functions, conformal maps to generalized quadrature domains, approximations for zeros of Hermite functions, qualities and bounds for elliptic integrals, P-symbols in Huen identities, iterative method for numerical integration of rational functions, a Taylor expansion theorem for any elliptic extension of the Askey-Wilson operator, and Ramanujan's symmetric theta functions in his lost notebook.
Their solutions are known as hypergeometric-type functions and to this class belong the Bessel, Airy, Weber, Whittaker, Gauss, Kummer, and Hermite functions, the classical orthogonal polynomials, among others.
3] Tina Marquardt and Andreas Ruffing, Modifying differential representations of Hermite functions and their impact on diffusion processes and Martingales, Dynam.
Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms.