Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.

The polynomials arise in:

• signal processing as Hermitian wavelets for wavelet transform analysis
• probability, such as the Edgeworth series, as well as in connection with Brownian motion;
• combinatorics, as an example of an Appell sequence, obeying the umbral calculus;
• numerical analysis as Gaussian quadrature;
• physics, where they give rise to the eigenstates of the quantum harmonic oscillator; and they also occur in some cases of the heat equation (when the term ${\displaystyle {\begin{aligned}xu_{x}\end{aligned}}}$ is present);
• systems theory in connection with nonlinear operations on Gaussian noise.
• random matrix theory in Gaussian ensembles.

Hermite polynomials were defined by Pierre-Simon Laplace in 1810,^[1][2] though in scarcely recognizable form, and studied in detail by Pafnuty Chebyshev in 1859.^[3] Chebyshev's work was overlooked, and they were named later after Charles Hermite, who wrote on the polynomials in 1864, describing them as new.^[4] They were consequently not new, although Hermite was the first to define the multidimensional polynomials.