\qquad
    \textnormal{or}
    \qquad
    0<\nu<1,\ \frac{\nu}{1-\nu+\nu^2}\le \mu<1.$$ For $\nu\le 0$ we can interpret as a random pushing caused by the jump of $x_{i-1}$, and as an independent jump of $x_i$ after the push (cf. ).

Setting $\nu=0$ in turns all the geometric-Bernoulli probabilities into the geometric ones with parameter $\mu$. This recovers the discrete time geometric PushTASEP as in . Thus, the $q=0$ degeneration of the $q$-Hahn PushTASEP can be viewed as a new nontrivial one-parameter extension of the geometric PushTASEP. Because $q$-moment formulas do not easily survive the $q\searrow0$ degeneration, here we do not pursue computations for this $q=0$ particle system.

A $q=0$ degeneration of the $q$-Hahn TASEP was introduced in [@derbyshev2012totally], [@povolotsky2015gen_tasep]. Its asymptotic behavior was studied recently in [@SaenzKnizelPetrov2018] via Schur measures.

Duality {#sub:duality}
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Like the $q$-Hahn TASEP, our $q$-Hahn PushTASEP satisfies a duality relation which we now describe. This is one of the main results of the present work.

Fix $N$ and $k$ and recall the spaces $\mathbb{X}^N$, $\mathbb{Y}^N$, and $\mathbb{Y}^{N}_k$ –. Let $\mathsf{P}^{\textnormal{PushTASEP}}_{q,\mu,\nu}$ denote the one-step Markov transition operator of the $q$-Hahn PushTASEP acting on $\mathbb{X}^{N}$ (the evolution of the $N$ rightmost particles under the $q$-Hahn PushTASEP is independent from the rest of the system). 