Riemann–Hilbert Problem and Matrix Biorthogonal Polynomials

Abstract

A. Recently the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, has been presented for matrix biorthogonal polynomials, in particular non-Abelian Hermite matrix biorthogonal polynomials in the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coe cients first order matrix polynomials. We will explore this discussion, present some achievements and consider some new examples of weights for matrix biorthogonal polynomials.

Hypergeometric expressions for type I Jacobi–Piñeiro orthogonal polynomials with arbitrary number of weights

“…Proof. In this context, we will leverage the fact that the polynomials defined in (11) can be obtained through Jacobi-Piñeiro as shown in (8) via the limit (12). In Theorem 2, we have established the following orthogonality conditions for the Jacobi-Piñeiro polynomials…”

“…This connection has been explored in various sources such as [9,14,15], see also [13]. Additionally, their significance in the context of Markov chains and random walks beyond birth and death has been established, as seen in [7,[10][11][12]. Notably, in both of these scenarios, type I polynomials play a pivotal role.…”

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

“…We begin this section stating a general theorem on the Riemann-Hilbert problem for the matrix Laguerre general weights. A preliminary version of this can be found in [54].…”

Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IV

“…We begin this section stating a general theorem on Riemann-Hilbert problem for the Laguerre general weights. A preliminary version of this can be found in [15].…”