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Gamma-ray bursts (GRBs) are the most intense explosions in the universe. GRBswith extended emission (GRB EE) constitute a small subclass of GRBs. GRB EE aredivided into EE-I GRBs and EE-II GRBs, according to the Amati empiricalrelationship rather than duration. We test here if these two types of GRB havedifferent origins based on their luminosity function (and formation rate).Therefore, we use Lynden-Bell's c^- method to investigate the LF and FR of GRBswith EE without any assumption. We calculate the formation rate of two types ofGRBs. 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) = 0.03Gpc^{-3}yr^{-1} for some EE-I GRBs and \rho (0) = 0.32 Gpc^{-3}yr^{-1} forEE-II GRBs. Based on these results, we provide a new evidence that the originsof 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-IIGRB bursts may come from other processes that are unrelated to the SFR. Ourfindings indicate that the GRB with EE could have multiple production channels.

New evidence of multiple channels for the origin of gamma-ray bursts with extended emission