Hermite polynomial

In mathematics and physics, Hermite polynomials form a well-known class of orthogonal polynomials. In quantum mechanics they appear as eigenfunctions of the harmonic oscillator and in numerical analysis they play a role in Gauss-Hermite quadrature. The functions are named after the French mathematician Charles Hermite (1822–1901).

The Hermite polynomials Hn(x) are orthogonal in the sense of the following inner product:

\left(H_{n'}, H_{n}\right)\equiv \int_{-\infty}^\infty H_{n'}(x)H_n(x)\; e^{-x^2}\, \mathrm{d}x = \delta_{n'n}\, h_n. $$ That is, the polynomials are defined on the full real axis and have weight w(x) = exp(&minus;x&sup2;). Their orthogonality is expressed by the appearance of the Kronecker delta &delta;n'n. The normalization constant is given by

h_n = \left(\frac{1}{\pi}\right)^{1/4}\, \frac{1}{\sqrt{2^n\,n!}}. $$