Orthogonal polynomials, Legendre
In summary, the first few Legendre polynomials are
\begin{equation*}
L_0(x) =1,
\end{equation*}
\begin{equation*}
L_1(x) = x,
\end{equation*}
\begin{equation*}
L_2(x) = (3x^2-1)/2,
\end{equation*}
\begin{equation*}
L_3(x) = (5x^3-3x)/2,
\end{equation*}
and
\begin{equation*}
L_4(x) = (35x^4-30x^2+3)/8.
\end{equation*}