  We consider the q-Hahn TASEP which is a three-parameter family of discrete
time interacting particle systems. The particles jump to the right
independently according to a certain q-Binomial distribution with parallel
updates. It is a generalization of the discrete time q-TASEP which is the
q-deformed totally asymmetric simple exclusion process (TASEP) on Z for q in
[0,1). For step initial condition, we prove that the current fluctuation of
q-Hahn TASEP at time t is of order $t^{1/3}$ and asymptotically distributed as
the GUE Tracy-Widom distribution. We verify the KPZ scaling theory conjecture
for the q-Hahn TASEP.
