---
title: "Hermite polynomials"
date: 2021-09-08
categories:
- Mathematics
- Statistics
layout: "concept"
---

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


## Physicists' 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:

$$\begin{aligned}
    \boxed{
        u'' - 2 x u' + 2 n u = 0
    }
\end{aligned}$$

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 *Rodrigues' formula*.
The first handful of Hermite polynomials are:

$$\begin{gathered}
    H_0(x) = 1
    \qquad \quad
    H_1(x) = 2 x
    \qquad \quad
    H_2(x) = 4 x^2 - 2
    \\
    H_3(x) = 8 x^3 - 12 x
    \qquad \quad
    H_4(x) = 16 x^4 - 48 x^2 + 12
\end{gathered}$$

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:

$$\begin{aligned}
    \boxed{
        \dvn{k}{}{x}H_n(x)
        = 2^k \frac{n!}{(n - k)!} H_{n - k}(x)
    }
\end{aligned}$$

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](/know/concept/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 $H_n$:

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