We introduce the q-Hahn asymmetric exclusion process, which is a two-sided generalization of the
q-Hahn TASEP (first introduced
here). We perform asymptotics, and observe that the density profile of the rarefaction fan is discontinuous when particles jump in both directions. The positions of particles fluctuate according to the KPZ scaling theory in the bulk. Surprisingly, we also prove that in the partially asymmetric case,
the location of the first particle also enjoys cube-root fluctuations which follow Tracy-Widom GUE
statistics.