utilize the duality to obtain contour integral formulas for the $q$-moments of this process.

$q$-Hahn PushTASEP, duality, and contour integrals {#sec:q_hahn_push}
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Definition and nonnegativity {#sub:def_nonnegativity}
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The $q$-Hahn PushTASEP depends on the main “quantization” parameter $q\in(0,1)$, and on two parameters $\mu,\nu$ in the following range: $$\label{mu_nu_parameters_for_qHahn}
    0<\mu<1,\qquad -1<\nu\le \min\{\mu,\sqrt q\}.$$ The $q$-Hahn PushTASEP is a discrete time Markov process on particle configurations in $\mathbb{Z}$ (with at most one particle per site) which have a rightmost particle: $$\vec x(t)=(x_1(t)> x_2(t)> \ldots).$$ At each discrete time moment, particles in the $q$-Hahn PushTASEP may jump to the left. The update $\vec x(t)\to \vec x(t+1)$ is performed according to the following procedure (the distributions $\varphi$ and $\psi$ are defined by and , respectively):

1.  The first particle $x_1$ jumps to the left by $\ell\in \mathbb{Z}_{\ge0}$, where $\ell$ is drawn from the distribution $\varphi_{q,\mu,\nu}(\ell\mid \infty)$.

2.  Consecutively for $i=2,3,\ldots $, given the movement of the $(i-1)$-st particle $x_{i-1}(t)\to x_{i-1}(t+1)=x_{i-1}(t)-\ell$, and the gap $g=x_{i-1}(t)-x_{i}(t)-1$ before this movement, the location of the $i$-th particle is updated as $x_i(t)\to x_{i}(t+1)=x_{i}(t)-L$, $L\in \mathbb{Z}_{\ge0}$, with probability $$\label{P_ell_g_update_probability_definition}
                \mathbf{P}_{\ell, g}(L):=
                \sum_{p=0}^{\min\{\ell, L\}}
                \varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)
                \,
                \psi_{q, \nu \mu^{-1} q^{p}, \nu q^{g}, \nu^{2} q^{g+p}}(L-p).$$ For consistency 