= Prolongations of the projective vector fields on the line
For $u_j=u^{(j)}$, the $n$th <prolongations of a vector field>[prolongations] of the translation, dilation, and special projective generators acting on the independent variable are
$$
\begin{aligned}
\operatorname{pr}^{(n)}(\partial_x)
&=\partial_x,\\
\operatorname{pr}^{(n)}(x\partial_x)
&=x\partial_x-\sum_{j=1}^n j u_j\partial_{u_j},\\
\operatorname{pr}^{(n)}(x^2\partial_x)
&=x^2\partial_x-\sum_{j=1}^n
\bigl(2jx u_j+j(j-1)u_{j-1}\bigr)\partial_{u_j}.
\end{aligned}
$$
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