Hermite Polynomials

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.

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 It might be outdated or ideologically biased.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 H_n (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

H_n+1 (x ) – 2xH_n (x ) + 2nH_n-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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.