NATO Science Series II. Mathematics, Physics and Chemistry vol:30 pages:23-59
NATO Advanced Study Institute on Special Functions 2000: Current Perspective and Future Directions location:Arizona State University date:May 29-June 9, 2000
In the early nineties, Fokas, Its and Kitaev observed that there is
a natural Riemann-Hilbert problem (for 2x2 matrix functions) associated which a system of orthogonal polynomials.
This Riemann-Hilbert problem was later used by Deift et al. and Bleher and Its
to obtain interesting results on orthogonal polynomials, in particular
strong asymptotics which hold uniformly in the complex plane. In this
paper we will show that a similar Riemann-Hilbert problem (for (r+1)x(r+1) matrix functions) is associated with
multiple orthogonal polynomials. We show how this helps in understanding
the relation between two types of multiple orthogonal polynomials and
the higher order recurrence relations for these polynomials. Finally we
indicate how an extremal problem for vector potentials is important for the normalization
of the Riemann-Hilbert problem. This extremal problem also describes the zero behavior of the multiple orthogonal polynomials.