Scalar and matrix Riemann-Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight (Q1885443)

An important development in the asymptotic theory of orthogonal polynomials is the introduction of methods based on matrix Riemann-Hilbert problems by \textit{P. Deift} et al. in [Comm. Pure Appl. Math. 52, 1335--1425, (1999; Zbl 0944.42013), 1491--1552 (1999; Zbl 1026.42024)] and \textit{P. Bleher} and \textit{A. Its} [Ann. Math. (2) 150, 185--266 (1999; Zbl 0956.42014)]. These papers were motivated by universality questions from random matrix theory. Boundary value problems on Riemann surfaces appeared in earlier works of \textit{J. Nuttall} [J. Approx. Theory 42, 299--386 (1984; Zbl 0565.41015)] and Stahl on Padé approximation. These boundary value problems can be considered as vector Riemann-Hilbert problems in contrast to the more recent matrix Riemann-Hilbert problems. In the present paper, the authors explain both methods and its relation to the strong asymptotics of Padé approximants to Markov functions. The boundary value problem on a Riemann surface is presented with some new ideas due to \textit{S. P. Suetin} [Mat. Sb. 191, No. 9, 81--114 (2000; Zbl 0980.41015)]. The matrix Riemann-Hilbert analysis is illustrated by a model problem of polynomials that are orthogonal on \([-1,1]\) with respect to a complex-valued weight function with square-root singularities at both end-points. These special weight functions have the advantage that the end-point analysis becomes trivial, which results in an easier and transparent exposition of the method.