We introduce two new exactly solvable (stochastic) interacting particle systems which are discrete time versions of q-TASEP. We call these geometric and Bernoulli discrete time q-TASEP. We obtain concise formulas for expectations of a large enough class of observables of the systems to completely characterize their fixed time distributions when started from step initial condition. We then extract Fredholm determinant formulas for the marginal distribution of the location of any given particle. Underlying this work is the fact that these expectations solve closed systems of difference equations which can be rewritten as free evolution equations with k-1 two-body boundary conditions -- discrete q-deformed versions of the quantum delta Bose gas. These can be solved via a nested contour integral ansatz. The same solutions also arise in the study of Macdonald processes, and we show how the systems of equations our expectations solve are equivalent to certain commutation relations involving the Macdonald first difference operator.

28 pages, 2 figures