ec 1 99 9 RIEMANN – HILBERT PROBLEM AND THE DISCRETE BESSEL KERNEL

We use discrete analogs of Riemann–Hilbert problem’s methods to derive the discrete Bessel kernel which describes the poissonized Plancherel measures for symmetric groups. To do this we define a discrete analog of 2 by 2 Riemann–Hilbert problems of special type. We also give an example, explicitly solvable in terms of classical special functions, when a discrete Riemann–Hilbert problem converges in a certain scaling limit to a conventional one; the example originates from the representation theory of the infinite symmetric group.