2025ApJ...978..160L

The merger of compact binary stars produces short gamma-ray bursts (sGRBs), involving channels such as neutron star-neutron star (BNS) and neutron star-black hole (NS-BH). The association between sGRB 170817A and gravitational wave GW170817 provides reliable evidence for the BNS channel. Some speculations suggest that sGRBs with extended emission (EE) may represent another distinct population. The offset is the distance between the GRB sky localization and the host galaxy center. We compared the offset distributions of these two types of samples (46 sGRBs with EE and 9 without EE samples) and found that they follow the same distribution. Utilizing nonparametric methods, we examined the luminosity function and formation rate of sGRBs without any assumptions. The luminosity function can be described as $\psi ({L}_{0})\propto {L}_{0}^{-0.12\pm 0.01}$ for ${L}_{0}\lt {L}_{0}^{b}$ ($\psi ({L}_{0})\propto {L}_{0}^{-0.73\pm 0.02}$ and for ${L}_{0}\gt {L}_{0}^{b}$) for sGRBs without EE and $\psi ({L}_{0})\propto {L}_{0}^{-0.13\pm 0.003}$ for ${L}_{0}\lt {L}_{0}^{b}$ ($\psi ({L}_{0})\propto {L}_{0}^{-0.61\pm 0.01}$ and for ${L}_{0}\gt {L}_{0}^{b}$) for sGRBs with EE. The formation rate is characterized as ρ(z) ∝ (1 + z)^{–3.04±0.10} for z < 1 and as ρ(z) ∝ (1 + z)^{–0.29±0.38} for 1 < z < 3 for sGRBs without EE, while for sGRBs with EE, it is ρ(z) ∝ (1 + z)^{–3.85±0.15} for z < 1 and ρ(z) ∝ (1 + z)^{–0.40±1.11} for 1 < z < 3. Our findings suggest no significant difference in the progenitors of sGRBs with and without EE when considered in terms of spatial offsets, formation rates, and luminosity function.