  Gamma-ray bursts (GRBs) are the most intense explosions in the universe. GRBs
with extended emission (GRB EE) constitute a small subclass of GRBs. GRB EE are
divided into EE-I GRBs and EE-II GRBs, according to the Amati empirical
relationship rather than duration. We test here if these two types of GRB have
different origins based on their luminosity function (and formation rate).
Therefore, we use Lynden-Bell's c^- method to investigate the LF and FR of GRBs
with EE without any assumption. We calculate the formation rate of two types of
GRBs. For EE-I GRBs, the fitting function can be written as \rho (z) \propto
{(1 + z)^{ - 0.34 \pm 0.04} for z < 2.39 and \rho (z) \propto {(1 + z)^{ - 2.34
\pm 0.24}} for z>2.39. The formation rate of EE-II can describe as \rho (z)
\propto {(1 + z)^{ - 1.05 \pm 1.10}} for z<0.43 and \rho (z) \propto {(1 + z)^{
- 8.44 \pm 1.10}} for z>0.43. The local formation rate are \rho (0) = 0.03
Gpc^{-3}yr^{-1} for some EE-I GRBs and \rho (0) = 0.32 Gpc^{-3}yr^{-1} for
EE-II GRBs. Based on these results, we provide a new evidence that the origins
of EE-I GRBs are different from EE-II GRBs from the perspective of event rate.
The EE-I GRB could be produced from the death of the massive star, but EE-II
GRB bursts may come from other processes that are unrelated to the SFR. Our
findings indicate that the GRB with EE could have multiple production channels.
