Hermite Polynomials

The Hermite polynomials are a classical sequence of orthogonal polynomials that arise across probability, combinatorics and physics, named after the French mathematician Charles Hermite, though earlier forms of them had already been studied by Pierre-Simon Laplace and Pafnuty Chebyshev. They form an orthogonal basis with respect to a Gaussian weight function, which is the reason they appear in probability theory in connection with the normal distribution, and they arise in physics as the eigenfunctions of the quantum harmonic oscillator, one of the few quantum systems that can be solved exactly. Two closely related but differently normalized conventions for the polynomials are in common use, the probabilists' Hermite polynomials and the physicists' Hermite polynomials, distinguished by which normalization of the Gaussian weight is used to define their orthogonality.