  We introduce new integrable exclusion and zero-range processes on the
one-dimensional lattice that generalize the $q$-Hahn TASEP and the $q$-Hahn
Boson (zero-range) process introduced in [Pov13] and further studied in
[Cor14], by allowing jumps in both directions. Owing to a Markov duality, we
prove moment formulas for the locations of particles in the exclusion process.
This leads to a Fredholm determinant formula that characterizes the
distribution of the location of any particle. We show that the model-dependent
constants that arise in the limit theorems predicted by the KPZ scaling theory
are recovered by a steepest descent analysis of the Fredholm determinant. For
some choice of the parameters, our model specializes to the
multi-particle-asymmetric diffusion model introduced in [SW98]. In that case,
we make a precise asymptotic analysis that confirms KPZ universality
predictions. 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.
