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In this paper we consider the $q$-deformed totally asymmetric zero rangeprocess ($q$-TAZRP), also known as the $q$-boson (stochastic) particle system,on the ${\mathbb Z}$ lattice, such that the jump rate of a particle depends onthe site where it is on the lattice. We derive the transition probability foran $n$ particle process in Bethe ansatz form as a sum of $n!$ $n$-fold contourintegrals. Our result generalizes the transition probability formula byKorhonen and Lee for $q$-TAZRP with a homogeneous lattice, and our methodfollows the same approach as theirs.

The Transition Probability of the $q$-TAZRP ($q$-Bosons) with Inhomogeneous Jump Rates