of notation we will sometimes write $x_0=+\infty$ and $\mathbf{P}_{\ell,\infty}(L)=\varphi_{q,\mu,\nu}(L\mid \infty)$. See for an illustration.

![Update of the $i$-th particle given the movement of the $(i-1)$-st particle.[]{data-label="fig:push_update_pic"}](fig-pushTASEP.pdf){width=".7\textwidth"}

Let us make a number of comments concerning this definition:

1.  It is not obvious that the right-hand side of is nonnegative because for $\nu>0$ the expression $\varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)$ might be negative. We prove the nonnegativity of the $\mathbf{P}_{\ell,g}$’s in below.

2.  The update probabilities $\mathbf{P}_{\ell,g}$ are given by a complicated expression involving $q$-Pochhammer symbols. In below we discuss a number of previously studied PushTASEP like processes which arise as degenerations of the $q$-Hahn PushTASEP. In particular, under these degenerations the $\mathbf{P}_{\ell,g}$’s simplify.

3.  One can check that $\mathbf{P}_{\ell,g}(L)=0$ unless $L\ge \ell-g$. (Indeed, this is because $\varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)=0$ unless $\ell-p\le g$.) In words, if the previous jumping distance $\ell$ is greater than the gap between $x_{i-1}$ and $x_i$, then the particle $x_i$ is deterministically pushed to the left. Therefore, update rule preserves the order of the particles.

4.  If $\ell>g$ and $\mu\nu$ equals $q^y$ for a positive integer $y$, the denominator in $\varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)$ may vanish. However, we can 