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

Abstract

In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights—which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem, are derived. An explicit and general example is presented to illustrate the theoretical results of the work. The non-Abelian extensions of a family of discrete Painlevé IV equations are discussed.

Two variable Freud orthogonal polynomials and matrix Painlevé-type difference equations

“…In [9,15] the matrix extension of dPI was first derived using the Riemann-Hilbert problem for the theory of matrix orthogonal polynomials. This has been extended further to alt-dPI, dPII and dPIV, see [4,5,6,10]. Matrix Painlevé systems have been also studied in [7,8].…”