[Paper Review] Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form

Paul Terwilliger|ArXiv.org|Apr 6, 2003

graph theory and CDMA systems31 references8 citations

TL;DR

This paper introduces two canonical forms—TD-D and LB-UB—for Leonard pairs, which are pairs of linear transformations each tridiagonal with respect to an eigenbasis of the other. The key contribution is a characterization of when tridiagonal-diagonal or lower/upper bidiagonal matrices form a Leonard pair, linking them to q-Racah polynomials via explicit parameterizations and hypergeometric series identities.

ABSTRACT

Let $\K$ denote a field and let $V$ denote a vector space over $\K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V o V$ and $B:V o V$ which satisfy both (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $B$ is diagonal; (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $B$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We introduce two canonical forms for Leonard pairs. We call these the TD-D canonical form and the LB-UB canonical form. In the TD-D canonical form the Leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix, subject to a certain normalization. In the LB-UB canonical form the Leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix, subject to a certain normalization. We describe the two canonical forms in detail. As an application we obtain the following results. Given square matrices $A,B$ over $\K$, with $A$ tridiagonal and $B$ diagonal, we display a necessary and sufficient condition for $A,B$ to represent a Leonard pair. Given square matrices $A,B$ over $\K$, with $A$ lower bidiagonal and $B$ upper bidiagonal, we display a necessary and sufficient condition for $A,B$ to represent a Leonard pair. We briefly discuss how Leonard pairs correspond to the $q$-Racah polynomials and some related polynomials in the Askey scheme. We present some open problems concerning Leonard pairs.

Motivation & Objective

• To define and characterize two canonical forms—TD-D and LB-UB—for Leonard pairs in finite-dimensional vector spaces over a field K.
• To provide necessary and sufficient conditions for a pair of matrices (one tridiagonal, one diagonal, or one lower/upper bidiagonal) to form a Leonard pair.
• To establish a correspondence between Leonard pairs and q-Racah polynomials in the Askey scheme via explicit parameterizations.
• To offer a framework for identifying and classifying Leonard pairs through matrix normalization and structural constraints.

Proposed method

• Introduce the TD-D canonical form: represent a Leonard pair as an irreducible tridiagonal matrix and a diagonal matrix under a specific normalization.
• Define the LB-UB canonical form: represent the pair as a lower bidiagonal and an upper bidiagonal matrix, again under normalization.
• Use the structure of eigenbases and matrix representations to derive recurrence relations and parameter constraints for the entries of the matrices.
• Employ basic hypergeometric series $_4\phi_3$ to express the entries of the transition matrices between eigenbases.
• Derive explicit formulas for the eigenvalues $\theta_i$, $\theta^*_j$ and the transition coefficients $k_j$, $k^*_j$ in terms of parameters $r_1, r_2, s, s^*, q, d$.
• Establish that the transition matrix entries are q-Racah polynomials in the eigenvalues, linking the algebraic structure to orthogonal polynomials.

Experimental results

Research questions

• RQ1 Under what conditions do a tridiagonal matrix A and a diagonal matrix B form a Leonard pair?
• RQ2 When does a lower bidiagonal matrix A and an upper bidiagonal matrix B form a Leonard pair?
• RQ3 How can the entries of the transition matrix between eigenbases be expressed in terms of q-Racah polynomials?
• RQ4 What is the precise normalization that defines the TD-D and LB-UB canonical forms?
• RQ5 How do the parameters $r_1, r_2, s, s^*, q, d$ determine the structure of the Leonard pair and its associated polynomials?

Key findings

• The TD-D canonical form exists and is unique up to normalization, with A tridiagonal and B diagonal, satisfying specific eigenvalue and transition coefficient constraints.
• The LB-UB canonical form exists and is unique under normalization, with A lower bidiagonal and B upper bidiagonal, providing an alternative matrix representation of the same Leonard pair.
• The transition matrix entries between eigenbases are given by the basic hypergeometric series $_4\phi_3$, which are identified as q-Racah polynomials in the eigenvalues.
• Explicit formulas for the eigenvalues $\theta_i$, $\theta^*_j$ and the transition coefficients $k_j$, $k^*_j$ are derived in terms of $q$-Pochhammer symbols and parameters $r_1, r_2, s, s^*, q, d$.
• The pair $A, A^*$ generates the matrix algebra $\mathrm{Mat}_{d+1}(\mathbb{K})$, confirming their algebraic independence and full generation of the ambient algebra.
• The structure of the Leonard pair corresponds precisely to the Askey-Wilson scheme, with the q-Racah polynomials arising naturally as the orthogonal polynomials associated with the eigenvalue sequences.