Riemann-Hilbert problems for multiple orthogonal polynomial

Abstract

Fokas, Its and Kitaev have shown in 1990-91 how orthogonal polynomials are related to a Riemann-Hilbert problem for a $2\times 2$ matrix valued analytic function with a matrix jump condition on the support of the weight function for the orthogonal polynomials. We will show how a similar Riemann-Hilbert problem for $(r+1)\times (r+1)$ matrix valued analytic functions gives rise to multiple orthogonal polynomials, which are defined by orthogonality conditions with respect to $r$ different weights. This Riemann-Hilbert approach is useful in obtaining a connection between type I and type II multiple orthogonal polynomials and some finite order recurrence relations. Furthermore, we indicate how this Riemann-Hilbert approach can be used for finding the asymptotics of these multiple orthogonal polynomials.