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