 Prolongations of the projective vector fields on the line

ID: prolongations-of-the-projective-vector-fields-on-the-line

For

u_{j} = u^{(j)}

, the

n

th prolongations of the translation, dilation, and special projective generators acting on the independent variable are

pr^{(n)} (\partial_{x}) pr^{(n)} (x \partial_{x}) pr^{(n)} (x^{2} \partial_{x}) = \partial_{x}, = x \partial_{x} - j = 1 \sum n j u_{j} \partial_{u_{j}}, = x^{2} \partial_{x} - j = 1 \sum n (2 j x u_{j} + j (j - 1) u_{j - 1}) \partial_{u_{j}} .

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