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 \(H_n\):

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


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