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.
In this case, we are able to explicitly compute the residue matrix at the simple pole at the origin of the structure matrix
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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.