Matrix-valued Cauchy bi-orthogonal polynomials and a novel noncommutative integrable lattice

TL;DR

This paper introduces matrix-valued Cauchy bi-orthogonal polynomials, derives their quasideterminant form, and reveals that their recurrence coefficients satisfy a new noncommutative integrable system characterized by fractional differential operators.

Contribution

It presents the first formulation of matrix-valued Cauchy bi-orthogonal polynomials and uncovers a novel noncommutative integrable lattice with a fractional differential Lax pair.

Findings

Quasideterminant expression for matrix-valued Cauchy bi-orthogonal polynomials

Recurrence coefficients satisfy a new noncommutative integrable system

Lax pair involves fractional differential operators with non-abelian variables

Abstract

Matrix-valued Cauchy bi-orthogonal polynomials were proposed in this paper, together with its quasideterminant expression. It is shown that the coefficients in four-term recurrence relation for matrix-valued Cauchy bi-orthogonal polynomials should satisfy a novel noncommutative integrable system, whose Lax pair is given by fractional differential operators with non-abelian variables.