A Riemann-Hilbert Approach to Asymptotic Analysis of Toeplitz+Hankel Determinants

Abstract

In this paper, we will formulate 4×4 Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported on an interval [a, b], 0 < a < b < 1. The distinguishing feature of this work is that in the formulation of the Riemann-Hilbert problem, no specific relationship is assumed between the Toeplitz and Hankel symbols. We will develop nonlinear steepest descent methods for analysing these problems in the case where the symbols are smooth (i.e., in the absence of Fisher-Hartwig singularities) and admit an analytic continuation in a neighborhood of the unit circle (if the symbol's support is the unit circle). We will finally introduce a model problem and will present its solution, requiring certain conditions on the ratio of Hankel and Toeplitz symbols. This, in turn, will allow us to find the asymptotics of the norms 𝘩ₙ of the corresponding orthogonal polynomials and, in fact, the large 𝑛 asymptotics of the polynomials themselves. We will explain how this solvable case is related to the recent operator-theoretic approach in [Basor E., Ehrhardt T., in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Oper. Theory Adv. Appl., Vol. 259, Birkhäuser/Springer, Cham, 2017, 125-154, arXiv:1603.00506] to Toeplitz+Hankel determinants. At the end, we will discuss the prospects of future work and outline several technical, as well as conceptual, issues that we are going to address next within the 4×4 Riemann-Hilbert framework introduced in this paper.