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
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.
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