Here $\varphi_{q,{\boldsymbol\upmu},{\boldsymbol\upnu}}$ is the $q$-beta-binomial distribution defined by . See for an illustration.

![$q$-Hahn TASEP and the update probabilities $\varphi_{q,{\boldsymbol\upmu},{\boldsymbol\upnu}}$ for $x_1$ and $x_2$.[]{data-label="fig:TASEP"}](fig-TASEP.pdf)

We now proceed to describe a Markov duality relation between the $q$-Hahn TASEP one-step transition operator and the $q$-Hahn Boson operator on a different space. Fix $N\ge1$ and define the $N$-particle space $$\label{space_X_N}
    \mathbb{X}^N:=
    \{\vec x:=(x_0,x_1,\ldots,x_N ), \ x_i\in \mathbb{Z},\
    +\infty=x_0>x_1>\ldots>x_N \}.$$ By $\mathsf{P}^{\textnormal{TASEP}}_{q,{\boldsymbol\upmu},{\boldsymbol\upnu}}$ denote the $q$-Hahn TASEP Markov transition operator acting on functions on $\mathbb{X}^N$. Note that by the very definition of the $q$-Hahn TASEP, the evolution of its $N$ rightmost particles is independent from the rest of the process.

Also define the Boson particle spaces $$\label{space_Y_N_y_Nk}
    \mathbb{Y}^N:=
    \{\vec y=(y_0,y_1,\ldots,y_N )\in\mathbb{Z}_{\ge0}^{N+1}\}
    ,
    \qquad
    \mathbb{Y}^N_k:=
    \{\vec y\in \mathbb{Y}^{N}\colon y_0+y_1+\ldots+y_N=k \}.$$ Let us define the $q$-Hahn Boson operator $\mathsf{P}^{\textnormal{Boson}}_{q,\alpha,{\boldsymbol\upnu}}$ acting on functions on $\mathbb{Y}^{N}$, where the parameter $\alpha$ is arbitrary. First, let $[\mathcal{A}_{q,\alpha,{\boldsymbol\upnu}}]_i$, $i=1,\ldots,N $, be a local operator acting only on the $y_i$ and $y_{i-1}$ coordinates of functions $f$ as $$[\mathcal{A}_{q,\alpha,{\boldsymbol\upnu}}]_i
    f(\vec y)
    =
    \sum_{s_i=0}^{y_i}
    \varphi_{q,\alpha,{\boldsymbol\upnu}}(s_i\mid y_i)\,
    f(y_0,y_1,\ldots,y_{i-1}+s_i,y_i-s_i,\ldots,y_N).$$ Define the $q$-Hahn Boson operator by its action $$\mathsf{P}^{\textnormal{Boson}}_{q,\alpha,{\boldsymbol\upnu}}
    f(\vec y)
    :=
    [\mathcal{A}_{q,\alpha,{\boldsymbol\upnu}}]_N
    \ldots
    [\mathcal{A}_{q,\alpha,{\boldsymbol\upnu}}]_1
    f(\vec y).$$ The order of operators is important and corresponds to parallel update: 