Orthogonal polynomials in several variables. I

TL;DR

This paper introduces classes of orthogonal polynomials in multiple non-commuting variables, focusing on their recurrence relations, Christoffel-Darboux formulas, and Jacobi matrices, extending classical concepts to a non-commutative setting.

Contribution

It develops a framework for orthogonal polynomials in several non-commuting variables, including recurrence relations and spectral analysis tools.

Findings

Defined non-commutative orthogonal polynomials

Derived recurrence equations and Christoffel-Darboux formulas

Constructed Jacobi type matrices for these polynomials

Abstract

In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations for these polynomials, Christoffel-Darboux formulas, and Jacobi type matrices.