Riemann-Hilbert problems for multiple orthogonal polynomial
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.