expected value of the duality function evolves over time.

The duality relation was guessed from the contour integral formulas () which generalize those known for the geometric $q$-pushTASEP (with step initial data). It was not a priori clear that the guessed formulas encoded expectations for any particle system. However, we discovered the $q$-Hahn pushTASEP introduced here satisfies both the duality (which is a result for general initial data) and the contour integral formulas (again, for step initial data).

Here we directly verify that the $q$-Hahn PushTASEP defined in satisfies . The proof of the duality relation occupies the rest of this subsection and is based on which we prove in the next .

First, let us write out for fixed $\vec x=(x_1,\ldots,x_N )\in \mathbb{X}^{N}$ and $\vec y=(y_0,y_1,\ldots,y_N )\in \mathbb{Y}^{N}_{k}$. We need to show that $$\label{push_duality_formulation_equation_as_sums}
    \mu^k
    \sum_{\vec x'\in \mathbb{X}^{N}}
    \sum_{\vec y'\in \mathbb{Y}^{N}_k}
    \mathsf{P}^{\textnormal{PushTASEP}}_{q,\mu,\nu}(\vec x,\vec x')
    \mathfrak{H}_k (\vec x',\vec y')
    \mathsf{P}^{\textnormal{Boson}}_{q,q/\mu,\nu}(\vec y,\vec y')
    =
    \nu^k
    \sum_{\vec y''\in \mathbb{Y}^{N}_k}
    \mathfrak{H}_k (\vec x,\vec y'')
    \mathsf{P}^{\textnormal{Boson}}_{q,q/\nu,\nu}(\vec y,\vec y'').$$ If $y_0>0$, both sides of vanish because $\mathsf{P}^{\textnormal{Boson}}_{q,q/\nu,\nu}(\vec y,\vec y')$ is nonzero only when $y_0'\ge y_0$, and we use the definition of $\mathfrak{H}$ . Thus, we can and will assume that $y_0=0$. Continuing, we further expand 