# Hermite Polynomials

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## Hermite polynomials

[er′mēt ‚päl·ə′nō·mē·əlz]## 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, *H*_{0} = 1, *H*_{1} = 2*x*, *H*_{2} = 4*x*^{2} – 2, *H*_{3} = 8*x*^{3} – 12*x*, and *H*_{4} = 16*x*^{4} – 48*x*^{2} + 12.

Hermite polynomials are orthogonal on the entire *x*-axis with respect to the weight function

*e ^{–x2}*

(*see*ORTHOGONAL POLYNOMIAL). They satisfy the differential equation

y″ – *2xy′ + 2ny* = 0

and the recursion formulas

*H _{n+1}* (

*x*) – 2

*xH*(

_{n}*x*) + 2

*nH*(

_{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}*–x ^{2}/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.