We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational.  In a number of classical special cases the recurrence procedure is equivalent to the difference and a q-Painlevé equations.  Our approach is based on the formalism of discrete integrable operators and discrete Riemann-Hilbert problems.  This is a joint work with D. Boyarchenko.