Is there an expression for $\exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = $?

Question

Does an expression for $$ \exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = ? $$ exist? For j=1 we have the usual expression for translation and scaling $$ \exp\left( t \partial_z\right) f(z) = f(z+t) $$ and $$ \exp\left( t z \partial_z\right) f(z) = f(e^tz) $$ Does similar expressions exist for higher order derivatives? How about if we restrict attention to $ f(z) = z^n $?