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 nn, and the Hermite polynomials Hn(x)H_n(x) are its eigenfunctions u(x)u(x), subject to the boundary condition that uu grows at most polynomially, in which case the eigenvalues nn are non-negative integers:

u2xu+2nu=0\begin{aligned} \boxed{ u'' - 2 x u' + 2 n u = 0 } \end{aligned}

The nnth-order Hermite polynomial Hn(x)H_n(x) is therefore as follows, according to physicists:

Hn(x)=(1)nexp(x2)dndxnexp(x2)=(2xddx)n1\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:

H0(x)=1H1(x)=2xH2(x)=4x22H3(x)=8x312xH4(x)=16x448x2+12\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 HnH_n can be computed quickly using the following recurrence relation:

Hn+1(x)=2xHn(x)2nHn1(x)\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:

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

Importantly, all HnH_n are orthogonal with respect to the weight function w(x)exp(x2)w(x) \equiv \exp(- x^2):

Hn|wHmHn(x)Hm(x)w(x)dx=π2nn!δnm\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 δnm\delta_{nm} is the Kronecker delta. Finally, they form a basis in the Hilbert space of all functions f(x)f(x) for which f|wf\Inprod{f}{w f} is finite. This means that every such ff can be expanded in HnH_n:

f(x)=n=0anHn(x)=n=0Hn|wfHn|wHnHn(x)\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}