not clear how to extend the PushTASEP analogously outside the Macdonald hierarchy. In this paper we achieve this. That is, we introduce a 3-parameter $q$-Hahn generalization of the PushTASEP (), adding a new parameter to the $q$-PushTASEP with $q$-geometric jumps discovered in [@MatveevPetrov2014]. This is akin to the $q$-Hahn extension of the discrete time $q$-TASEP [@BorodinCorwin2013discrete] found in [@Povolotsky2013] and studied in [@Corwin2014qmunu]. Our pushing system enjoys a variant of Markov duality (see ) similar to the one studied earlier for the continuous time $q$-PushTASEP [@CorwinPetrov2013], and generalizes the contour integral formulas from that setting, too (see ). The system itself is not so simple (in particular, its transition probabilities are expressed through the basic hypergeometric function $_4\phi_3$), and our search for it was informed by the desire to extend the duality and contour integral formulas away from the Macdonald context. Even once this generalized system is introduced (see ), it is not so simple to prove the duality, as it requires some interesting $q$-identities (the central statement is which we prove in ).

The $q$-Hahn TASEP revealed some interesting new systems as its limits — in particular, the beta polymer (or random walk in random environment) considered in [@CorwinBarraquand2015Beta] 