Abstract. We consider the large-N asymptotics of a system of discrete orthogonal polynomials on an infinite regular lattice of mesh 1 N , with weight e , where V (x) is a real analytic function with sufficient growth at infinity. The proof is based on formulation of an interpolation problem for discrete orthogonal polynomials, which can be converted to a Riemann-Hilbert problem, and steepest de...

Let Σ be a bordered Riemann surface with genus g and m boundary components. Let {γz}z∈∂Σ be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Then there exists a holomorphic function f(z) on Σ smooth up to the boundary with at most 2g +m− 1 zeros on Σ such that f(z) ∈ γz for every z ∈ ∂Σ.

We derive a Riemann–Hilbert problem satisfied by the Baker-Akhiezer function for the finite-gap solutions of the Korteweg-de Vries (KdV) equation. As usual for Riemann-Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears in an explicit, linear and computable way. We make use of recent advances in the nume...

A new, numerical framework for the approximation of solutions to matrix-valued Riemann–Hilbert problems is developed, based on a recent method for the homogeneous Painlevé II Riemann–Hilbert problem. We demonstrate its effectiveness by computing solutions to other Painlevé transcendents. An implementation in Mathematica is made available online.

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on [−1, 1]. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann-Hilbert problem for orthogonal polynomials. Using the steepest descent method of Deift and...

Angelesco systems of measures with Jacobi type weights are considered. For such systems, strong asymptotics for the related multiple orthogonal polynomials are found as well as the Szegő-type functions. In the procedure, an approach from Riemann-Hilbert problem plays a fundamental role.