Product-type non-commutative polynomial states

TL;DR

This paper explores product-type states on non-commutative polynomials, illustrating their properties through examples and connecting them to universal products, thereby advancing understanding of non-commutative orthogonal polynomial states.

Contribution

It introduces a new notion of product-type states on polynomials that encompasses all non-commutative universal products and demonstrates their properties through examples.

Findings

Defines a new class of product-type states on non-commutative polynomials

Shows these states include all non-commutative universal products

Provides examples illustrating the properties of these states

Abstract

In math/0702157, arXiv:0712.4185, we investigated monic multivariate non-commutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of examples, demonstrating what these general results look like for the most important states on non-commutative polynomials, namely for various product states. In particular, we introduce a notion of a product-type state on polynomials, which covers all the non-commutative universal products and excludes some other familiar non-commutative products, and which guarantees a number of nice properties for the corresponding polynomials.