Other orthogonal polynomials, Hermite polynomials
A typical example is again the solution of Schrodinger's
equation, but this time with a harmonic oscillator potential.
The first few polynomials are
$$
\begin{equation*}
H_0(x)=1,
\end{equation*}
$$
$$
\begin{equation*}
H_1(x)=2x,
\end{equation*}
$$
$$
\begin{equation*}
H_2(x)=4x^2-2,
\end{equation*}
$$
$$
\begin{equation*}
H_3(x)=8x^3-12,
\end{equation*}
$$
and
$$
\begin{equation*}
H_4(x)=16x^4-48x^2+12.
\end{equation*}
$$
They fulfil the orthogonality relation
$$
\begin{equation*}
\int_{-\infty}^{\infty}e^{-x^2}H_n(x)^2dx=2^nn!\sqrt{\pi},
\end{equation*}
$$
and the recursion relation
$$
\begin{equation*}
H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x).
\end{equation*}
$$