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Riemann‐Hilbert problem and matrix discrete Painlevé II systems
Author(s) -
CassatellaContra Giovanni A.,
Mañas Manuel
Publication year - 2019
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12277
Subject(s) - mathematics , matrix (chemical analysis) , matrix function , pure mathematics , biorthogonal system , ordinary differential equation , matrix differential equation , hilbert matrix , nonnegative matrix , pascal matrix , cauchy distribution , mathematical analysis , algebra over a field , differential equation , symmetric matrix , eigenvalues and eigenvectors , hilbert space , materials science , composite material , physics , wavelet transform , reproducing kernel hilbert space , quantum mechanics , artificial intelligence , computer science , projective hilbert space , wavelet
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.