Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Abstract

Matrix Szegő biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szegő polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szegő matrix and the associated Szegő recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szegő polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.

Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IV

“…De nition 3. Motivated by (19) and (21) we introduce two linear operators L and R , acting on the linear space of polynomials C N×N [z] as follows L (P) := zP + P a L (z) + Pb L (z), R (P) := zP + a R (z)P + b R (z)P.…”

“…in the whole space of parameters except possibly for algebraic subvarieties. The situation was considered in [19] for the matrix extension of the Szegő polynomials in the unit circle and corresponding non-Abelian versions discrete Painlevé II equations.…”

Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

“…It was found that the singularity confinement holds generically, i.e., in the whole space of parameters except possibly for algebraic subvarieties. This situation was considered in [51] for the matrix extension of the Szegő polynomials in the unit circle and corresponding non-Abelian versions of discrete Painlevé II equations.…”

Ladder relations for a class of matrix valued orthogonal polynomials

“…This Riemann–Hilbert formulation is a powerful methodology to obtain algebraic and differential identities for MVOPs, as well as for the functions of the second kind, and it has been extensively used in the last few years, we refer the reader to Refs. 25–27 and to Refs. 28, 29 for matrix orthogonal polynomials of Hermite and Laguerre type on the real line or the positive half‐line.…”