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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:

$$\begin{aligned} H_n(x) &= (-1)^n \exp(x^2) \dvn{n}{}{x}\exp(- x^2) \\ &= \Big( 2 x - \dv{}{x}\Big)^n 1 \end{aligned}$$

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 of all functions $f(x)$ for which $\Inprod{f}{w f}$ is finite. This means that every such $f$ can be expanded in $H_n$:

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

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