introduces a pushing mechanism. It seems likely that duality and contour integral formulas can be carried over from the original stochastic higher spin model. On the other hand, to recover the $q$-Hahn PushTASEP one must find the right analytic continuation. We leave this for a later investigation.

We also note that it should be possible to produce a 4-parameter family of pushing systems by a different mechanism — via bijectivisation of the Yang-Baxter equation (see [@BufetovPetrovYB2017]) related to the spin $q$-Whittaker polynomials (introduced in [@BorodinWheelerSpinq]). This approach is developed in the upcoming work [@BufetovMucciconiPetrov2018], and we anticipate that the same 4-parameter family will come up in this manner.

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In we recall duality and contour integral formulas for the $q$-moments of the $q$-Hahn TASEP [@Povolotsky2013], [@Corwin2014qmunu], as some of these ingredients are used for the $q$-Hahn PushTASEP. In we introduce the $q$-Hahn PushTASEP, discuss its various degenerations, and prove duality and contour integral formulas. In we consider a beta polymer-like limit of the $q$-Hahn PushTASEP as $q\to1$, and write down moments of the resulting system in a contour integral form. We also provide a conjecture for the $q$-Laplace transform. Formulas pertaining to $q$-hypergeometric functions and associated probability 