Categories: Mathematics, Statistics.

# Hermite polynomials

The **Hermite polynomials** are a set of functions
that appear in physics and statistics,
although slightly different definitions are used in those fields.

## Physicistsâ€™ definition

The **Hermite equation** is an eigenvalue problem for $n$,
and the Hermite polynomials $H_n(x)$ are its eigenfunctions $u(x)$,
subject to the boundary condition that $u$ grows at most polynomially,
in which case the eigenvalues $n$ are non-negative integers:

The $n$th-order Hermite polynomial $H_n(x)$ is therefore as follows, according to physicists:

$\begin{aligned} H_n(x) &= (-1)^n \exp(x^2) \dvn{n}{}{x}\exp(- x^2) \\ &= \Big( 2 x - \dv{}{x}\Big)^n 1 \end{aligned}$This form is known as a *Rodriguesâ€™ formula*.
The first handful of Hermite polynomials are:

And then more $H_n$ can be computed quickly using the following recurrence relation:

$\begin{aligned} \boxed{ H_{n + 1}(x) = 2 x H_n(x) - 2n H_{n-1}(x) } \end{aligned}$They (almost) form an *Appell sequence*,
meaning their derivatives are like so:

Importantly, all $H_n$ are orthogonal with respect to the weight function $w(x) \equiv \exp(- x^2)$:

$\begin{aligned} \boxed{ \Inprod{H_n}{w H_m} \equiv \int_{-\infty}^\infty H_n(x) \: H_m(x) \: w(x) \dd{x} = \sqrt{\pi} 2^n n! \: \delta_{nm} } \end{aligned}$Where $\delta_{nm}$ is the Kronecker delta. Finally, they form a basis in the Hilbert space of all functions $f(x)$ for which $\Inprod{f}{w f}$ is finite. This means that every such $f$ can be expanded in $H_n$:

$\begin{aligned} \boxed{ f(x) = \sum_{n = 0}^\infty a_n H_n(x) = \sum_{n = 0}^\infty \frac{\Inprod{H_n}{w f}}{\Inprod{H_n}{w H_n}} H_n(x) } \end{aligned}$