[Paper Review] Orthogonal polynomials in several non-commuting variables. II

T. Banks, T. Constantinescu|ArXiv.org|Dec 29, 2004

Mathematical functions and polynomials8 references3 citations

TL;DR

This paper develops a multivariable extension of classical orthogonal polynomial theory for non-commuting variables, introducing a matrix-based recursive framework that characterizes Hankel-type positive definite kernels via three-term recurrence relations. The key contribution is a complete structural description of such kernels through Jacobi coefficients derived from moments, with explicit formulas for the recurrence matrices in terms of moment parameters and lexicographic ordering.

ABSTRACT

In this paper we continue to investigate a certain class of Hankel-like positive definite kernels using their associated orthogonal polynomials. The main result of this paper is about the structure of this kind of kernels.

Motivation & Objective

• To characterize positive definite kernels on the free semigroup of N generators that satisfy a specific invariance property under word reversal and juxtaposition.
• To generalize classical one-variable orthogonal polynomial theory—particularly the three-term recurrence and moment-Jacobi coefficient correspondence—to the non-commutative, multivariable setting.
• To establish a complete algebraic and analytic structure for these kernels using orthonormal polynomials and their associated recurrence matrices.
• To describe the Jacobi coefficients of free products of orthogonal polynomials in non-commuting variables, extending known results from the commutative case.

Proposed method

• Uses the Gelfand-Naimark-Segal (GNS) construction to associate a Hilbert space with a strictly positive linear functional on non-commutative polynomials.
• Applies the Gram-Schmidt process with lexicographic ordering on the free semigroup to construct orthonormal polynomials indexed by words.
• Introduces matrix notation $\Phi_n = [\varphi_\sigma]_{|\sigma|=n}$ to represent families of orthonormal polynomials of fixed degree, enabling compact expression of recurrence relations.
• Derives a three-term recurrence relation $X_k\Phi_n = \Phi_{n+1}A_{n+1,k} + \Phi_n B_{n,k} + \Phi_{n-1}A^*_{n,k}$ for each generator $X_k$, with $A_{n,k}$ and $B_{n,k}$ matrices defined via moment parameters.
• Constructs explicit formulas for the matrices $A_{n,k}$ and $B_{n,k}$ using lexicographic word ordering and the depth of non-repeating blocks, ensuring $A_n = [A_{n,1}, \dots, A_{n,N}]$ is upper triangular and invertible with positive diagonal entries.
• Defines an admissible family of matrices $\mathcal{A}$ such that the recurrence holds for all $n \geq 0$, $k=1,\dots,N$, and proves the existence of such a structure via induction on word length.

Experimental results

Research questions

• RQ1 How can the classical moment-Jacobi coefficient correspondence be extended to orthogonal polynomials in several non-commuting variables?
• RQ2 What structural properties characterize positive definite kernels on the free semigroup satisfying the invariance $K(\alpha\sigma,\tau) = K(\sigma, I(\alpha)\tau)$?
• RQ3 How do the recurrence coefficients (Jacobi matrices) in the multivariable non-commutative setting relate to the moments of the underlying functional?
• RQ4 What is the form of the recurrence matrices $A_{n,k}$ and $B_{n,k}$ in terms of the moment parameters and word structure?
• RQ5 How do free products of orthogonal polynomials behave in this non-commutative framework, and what is their Jacobi coefficient structure?

Key findings

• The paper establishes a multivariable extension of the classical moment-Jacobi coefficient correspondence, showing that the Jacobi coefficients of non-commutative orthogonal polynomials are completely determined by the moments of the underlying functional.
• It proves that the recurrence matrices $A_{n,k}$ and $B_{n,k}$ are explicitly constructed from the moment parameters and the lexicographic ordering of words, with $A_n$ being upper triangular and invertible with strictly positive diagonal entries.
• The matrices $A_{n,k}$ and $B_{n,k}$ are defined via block-diagonal structures based on the number of leading identical letters in each word, with entries $a_{m,k}$ and $b_{m,k}$ from the one-variable orthogonal polynomial recurrence.
• For any $N$, the recurrence $X_k\Phi_n = \Phi_{n+1}A_{n+1,k} + \Phi_n B_{n,k} + \Phi_{n-1}A^*_{n,k}$ holds for all $n \geq 0$ and $k=1,\dots,N$, with the matrices defined via the depth of non-repeating blocks in words.
• The family $\mathcal{A} = \{A_{n,k}, B_{m,k} \mid n>0, m\geq 0, k=1,\dots,N\}$ is shown to be admissible, ensuring the recurrence structure is consistent and well-defined across all degrees.
• The construction generalizes to free products of orthogonal polynomials, with the Jacobi coefficients of the product being determined by the component coefficients through the same matrix recurrence framework.