The transition probability and the probability for the left-most particle's position of the q-TAZRP

We treat the $N$-particle ZRP whose jumping rates satisfy a certain condition. This condition is required to use the Bethe ansatz and the resulting model is the $q$-boson model that appeared in [J. Phys. A, \textbf{31} 6057--6071 (1998)] by Sasamoto and Wadati or the $q$-TAZRP in \textit{MacDonald processes} by Borodin and Corwin. We find the explicit formula of the transition probability of the $q$-TAZRP via the Bethe ansatz. By using the transition probability we find the probability distribution of the left-most particle's position at time $t$. To find the probability for the left-most particle's position we find a new identity corresponding to Tracy and Widom's identity for the ASEP in [Commun. Math. Phys., \textbf{279} 815--844 (2008)]. For the initial state that all particles occupy a single site, the probability distribution of the left-most particle's position at time $t$ is represented by the contour integral of a determinant.