Matrix biorthogonal polynomials, associated polynomials and functions of the second kind

TL;DR

This paper explores the relationships between matrix biorthogonal polynomials, associated polynomials, and second kind functions using quasideterminants, providing new formulas and a Poincaré-type theorem for matrix functions.

Contribution

It introduces novel connections and formulas linking matrix biorthogonal polynomials, associated polynomials, and second kind functions, expanding the theoretical framework.

Findings

New formulas connecting matrix polynomial families

A Poincaré-type theorem for matrix function ratios

Insights into the structure of matrix biorthogonal polynomials

Abstract

In this work the interplay between matrix biorthogonal polynomials with respect to a matrix of linear functionals, the $k$-th associated matrix polynomials and the second kind matrix functions, is studied in terms of quasideterminants. A sort of Poincar\'{e}'s theorem for the ratio of two consecutive matrix functions solutions of a linear difference equation is also presented. Some new formulas connecting these families of matrix functions are given.