The Hermite polynomials are a set of functions
that appear in physics and statistics,
although slightly different definitions are used in those fields.
The Hermite equation is an eigenvalue problem for ,
and the Hermite polynomials are its eigenfunctions ,
subject to the boundary condition that grows at most polynomially,
in which case the eigenvalues are non-negative integers:
The th-order Hermite polynomial
is therefore as follows, according to physicists:
This form is known as a Rodrigues’ formula.
The first handful of Hermite polynomials are:
And then more can be computed quickly
using the following recurrence relation:
They (almost) form an Appell sequence,
meaning their derivatives are like so:
Importantly, all are orthogonal with respect to the weight function :
Where is the Kronecker delta.
Finally, they form a basis in the Hilbert space
of all functions for which is finite.
This means that every such can be expanded in :