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

JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS (2022)

Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painleve IV

Amilcar Branquinho et al.

Summary: In this paper, the Riemann-Hilbert problem is used to study matrix biorthogonal polynomials associated with Laguerre type matrices. Differential systems for the fundamental matrix are derived and an explicit example is provided. The non-Abelian extensions of a family of discrete Painleve IV equations are also discussed.

MATHEMATICS (2022)

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Article Mathematics, Applied

Matrix biorthogonal polynomials: Eigenvalue problems and non-Abelian discrete Painleve equations A Riemann-Hilbert problem perspective

Amilcar Branquinho et al.

Summary: This paper utilizes the Riemann-Hilbert problem for matrix biorthogonal polynomials, finding Sylvester systems of differential equations for orthogonal polynomials and second kind functions. It demonstrates the use of Sylvester type differential Pearson equations for the matrix of weights, and explores various scenarios leading to eigenvalue problems for second order linear differential operators with matrix eigenvalues. The discussion covers non-Abelian Hermite biorthogonal polynomials, nonlinear matrix difference equations, and extensions of the discrete Painleve I equation.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (2021)

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Article Physics, Multidisciplinary