The paper sets up a Riemann-Hilbert problem for matrix Laguerre biorthogonal polynomials from a matrix Pearson equation, derives first- and second-order differential systems, and connects them to eigenvalue problems and matrix discrete Painlevé IV equations.
Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlev\'e IV
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abstract
In this paper the Riemann-Hilbert problem, with jump supported on a appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of 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. Related matrix eigenvalue problems for second order matrix differential operators and non-Abelian extensions of a family of discrete Painlev\'e IV equations are discussed.
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2019 1verdicts
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Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlev\'e IV
The paper sets up a Riemann-Hilbert problem for matrix Laguerre biorthogonal polynomials from a matrix Pearson equation, derives first- and second-order differential systems, and connects them to eigenvalue problems and matrix discrete Painlevé IV equations.