Chain models and the spectra of tridiagonal k-Toeplitz matrices

Chain models can be represented by a tridiagonal matrix with periodic entries along its diagonals. Eigenmodes of open chains are represented by spectra of such tridiagonal $k$-Toeplitz matrices, where $k$ represents length of the repeated unit, allowing for a maximum of $k$ distinct types of elements in the chain. We present an analysis that allows for generality in $k$ and values in $\mathbb{C}$ representing elements of the chain, including non-Hermitian systems. Numerical results of spectra of some special $k$-Toeplitz matrices are presented as a motivation. This is followed by analysis of a general tridiagonal $k$-Toeplitz matrix of increasing dimensions, beginning with 3-term recurrence relations between their characteristic polynomials involving a $k^{th}$ order coefficient polynomial, with the variables and coefficients in $\mathbb{C}$. The existence of limiting zeros for these polynomials and their convergence are established, and the conditioned $k^{th}$ order coefficient polynomial is shown to provide a continuous support for the limiting spectra representing modes of the chain. This analysis also includes the at most $2k$ eigenvalues outside this continuous set. It is shown that this continuous support can as well be derived using Widom's conditional theorems (and its recent extensions) for the existence of limiting spectra for block-Toeplitz operators, except in special cases. Numerical examples are used to graphically demonstrate theorems. As an addendum, we derive expressions for $O(k)$ computation of the determinant of tridiagonal $k$-Toeplitz matrices of any dimension

Cyclic tridiagonal pairs, higher order Onsager algebras and orthogonal polynomials

The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is identified as a higher-order generalization of the Onsager algebra. It can be viewed as a subalgebra of the q-Onsager algebra for a proper specialization at q the primitive 2Nth root of unity. Orthogonal polynomials beyond the Leonard duality are revisited in light of this framework. In particular, certain second-order Dunkl shift operators provide a realization of the divided polynomials at N=2 or q=i.Comment: 32 pages; v2: Appendices improved and extended, e.g. a proof of irreducibility is added; v3: version for Linear Algebra and its Applications, one assumption added in Appendix about eq. (A.2

Determinants of Block Tridiagonal Matrices

An identity is proven that evaluates the determinant of a block tridiagonal matrix with (or without) corners as the determinant of the associated transfer matrix (or a submatrix of it).Comment: 8 pages, final form. To appear on Linear Algebra and its Application

Tridiagonal realization of the anti-symmetric Gaussian β\beta-ensemble

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pr\"ufer phases of the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal transformation proof for both cases (Method III

Doubling (Dual) Hahn Polynomials: Classification and Applications

We classify all pairs of recurrence relations in which two Hahn or dual Hahn polynomials with different parameters appear. Such couples are referred to as (dual) Hahn doubles. The idea and interest comes from an example appearing in a finite oscillator model [Jafarov E.I., Stoilova N.I., Van der Jeugt J., J. Phys. A: Math. Theor. 44 (2011), 265203, 15 pages, arXiv:1101.5310]. Our classification shows there exist three dual Hahn doubles and four Hahn doubles. The same technique is then applied to Racah polynomials, yielding also four doubles. Each dual Hahn (Hahn, Racah) double gives rise to an explicit new set of symmetric orthogonal polynomials related to the Christoffel and Geronimus transformations. For each case, we also have an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. This extends the class of Sylvester-Kac matrices by remarkable new test matrices. We examine also the algebraic relations underlying the dual Hahn doubles, and discuss their usefulness for the construction of new finite oscillator models

A family of tridiagonal pairs and related symmetric functions

A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order $q-$difference equations which generalize the ones associated with the Askey-Wilson orthogonal polynomials with a discrete argument. Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples. The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given.Comment: 17 pages; LaTeX file with amssymb. v2: few minor changes, to appear in J.Phys.A; v3: Minor misprints, eq. (48) and orthogonality condition corrected compared to published versio