1-cocycles on the group of contactomorphisms on the supercircles S1|1 and S1|2 generalizing the Schwarzian derivative

We give an explicit construction of the $1$-cocycles of the group of contactomorphisms on the supercircle $S^{1|m}$ for $m=1,2$, with coefficients in the space of differential operators acting on $S^{1|m}$-tensor densities. We show that they satisfy properties similar to those of the super-Schwarzian derivative.