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.

## 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) \dv[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{ \dv[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{ \braket{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 $$\braket{f}{w f}$$ 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{\braket{H_n}{w f}}{\braket{H_n}{w H_n}} H_n(x) } \end{aligned}