  We give contour integral formulas for the generating function of the joint
distribution of the PushASEP on a ring. We obtained these formulas through a
rigorous treatment of Bethe Ansatz. The approach relies on residue computations
and controlling the location of the Bethe roots, which we achieve by partially
decoupling the Bethe equations through extending the system of equations.
Moreover, we are able to use our formulas to compute the asymptotic
fluctuations for the flat and step initial conditions at the relaxation time
scale.
