Matrix biorthogonal polynomials: Eigenvalue problems and non-Abelian discrete Painlevé equations

In this paper we use the Riemann–Hilbert problem, with jumps supported on appropriate curves in complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations orthogonal its second kind functions as well. For aim, type Pearson weights are shown be instrumental. Several applications given, order increasing complexity. First, a general discussion non-Abelian Hermite real line, understood those whose is solution equation coefficients first degree polynomials, given. All these applied possible scenarios leading eigenvalue problems linear operators eigenvalues. Nonlinear difference discussed next. Firstly, situation non relation (non trivial because commutativity features setting) recursion gotten. next case higher difficulty, two allowed equation, but simplified by considering only left equation. case, support measure an branch hyperbola. The fulfill extension alternate discrete Painlevé I Finally, given three characterizing weights. However, simplicity odd allowed. new more found.