\nu q^{\frac{\ell-g-1}2}
                & q^{\ell-g+1} & \nu/ \mu q^{\ell-g} & \nu & \nu^{2}q^{L} & \nu^{2}q^{\ell}
            \end{array}; q, \frac{q^{L+g+1}}{\mu}
        \right].
    \end{aligned}$$ In both cases and all the prefactors and the terms in the sums for $_8\phi_7$ are manifestly nonnegative under our conditions (for the terms in $_8\phi_7$ this follows from the definition of $_8\phi_7$ as a sum). This completes the proof.

\[rmk:sqrt\_q\_maybe\_a\_proof\_artefact\] The condition $\nu\le \sqrt{q}$ in , which was not present in the $q$-Hahn TASEP [@Corwin2014qmunu], [@CorwinPetrov2015], is essential for the nonnegativity of transition probabilities in our $q$-Hahn PushTASEP. Indeed, $$\mathbf{P}_{1,1}(1)=
        \frac{\mu  \left(\nu ^2-\nu +1\right)-\nu +q \left( 1+\nu ^2-(\mu +1) \nu\right)}{(1-\mu  \nu ) (1-q \nu ^2)},$$ and the numerator in this expression is negative for, say, $q=\frac{1}{4}$, $\mu=\frac{3}{4}$, and $\nu=\frac{2}{3}$.

Degenerations {#sub:degenerations}
-------------

The $q$-Hahn PushTASEP update probabilities $\mathbf{P}_{\ell,g}(L)$ are defined by rather complicated expressions . Here we discuss a number of their degenerations when the parameter $q=0$. These lead to some known and some new stochastic particles systems with pushing. In we discuss another type of degeneration where $q\to 1$ which also simplifies the form of the update probabilities.

### Known PushTASEPs

If we set $\nu=0$, the factor $\psi_{q, \nu \mu^{-1} q^{p}, \nu q^{g}, \nu^{2} q^{g+p}}(L-p)$ in simplifies to $$\psi_{q, 