Noncommutative Biorthogonal Polynomials

TL;DR

This paper introduces a new algebraic framework for noncommutative biorthogonal polynomials, deriving recurrence relations and extending Favard's theorem to this generalized setting.

Contribution

It combines noncommutative and biorthogonal polynomial theories into a unified algebraic approach, including new recurrence relations and a broad extension of Favard's theorem.

Findings

Established an algebraic definition of noncommutative biorthogonal polynomials

Derived recurrence relations for certain biorthogonal polynomial types

Extended Favard's theorem to the noncommutative biorthogonal context

Abstract

The idea of orthogonal polynomials has been generalized in two ways to achieve new types of polynomials: noncommutative orthogonal polynomials and biorthogonal polynomials. This paper brings these two different generalizations together to present a completely algebraic definition of noncommutative biorthogonal polynomials. It then goes on to obtain recurrence relations for some types of biorthogonal polynomials and concludes with a broad extension of Favard's theorem.