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 H_n(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(−x²). Their orthogonality is expressed by the appearance of the Kronecker delta δ_n'n. The normalization constant is given by

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

References

Abromowitz and Stegun Chapter 22