Discrete Riemann-Hilbert problem and interpolation of entire functions

We consider two problems in complex analysis which were developed in Ufa in 1970s years. These are a Riemann-Hilbert problem about jump of a piecewise-analytic function on a contour and a problem of interpolation of entire functions on a countable set in the complex plane. A progress in recent years led to comprehension that they have much common in subject. The first problem arrives as an equivalent of the inverse scattering problem applied for integrating nonlinear differential equations of mathematical physics. The second problem is a natural generalization of Lagrange formula for polynomial with given values on a finite set of points. It is shown that both problems can be united by generalization of the Riemann-Hilbert problem on a case of “discrete contour”, where a “jump” of analytic function takes place. This formulation of the discrete matrix Riemann problem can be applied now for various problems of exactly solvable difference equations as well as estimates of spectrum of random matrices. In the paper we show how the discrete matrix Riemann-Hilbert problem provides a way to integrate nonlinear difference equations such as a discrete Painlevé equation. On the other hand, it is shown how assignment of residues to meromorphic matrix functions is effectively reduced to an interpolation problem of entire functions on a countable set in $\mathbb{C}$ with the only accumulation point at infinity. Other application of discrete matrix Riemann-Hilbert problem includes calculation of Fredholm determinants emerging in combinatorics and group representation theory.