Orthogonal Polynomials from Hermitian Matrices II

This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended $\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schr\"odinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the $q$-Meixner ($q$-Charlier) polynomials do not form a complete basis of the $\ell^2$ Hilbert space, based on the fact that the dual $q$-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the $q$-Meixner polynomials is obtained by constructing the duals of the dual $q$-Meixner polynomials which require the two component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big $q$-Jacobi family and their duals and $q$-Meixner ($q$-Charlier) polynomials.Comment: 65 pages. Comments, references and table of contents are added. To appear in J.Math.Phy

Quantum dimensions and their non-Archimedean degenerations

We derive explicit dimension formulas for irreducible $M_F$-spherical $K_F$-representations where $K_F$ is the maximal compact subgroup of the general linear group $GL(d,F)$ over a local field $F$ and $M_F$ is a closed subgroup of $K_F$ such that $K_F/M_F$ realizes the Grassmannian of $n$-dimensional $F$-subspaces of $F^d$. We explore the fact that $(K_F,M_F)$ is a Gelfand pair whose associated zonal spherical functions identify with various degenerations of the multivariable little $q$-Jacobi polynomials. As a result, we are led to consider generalized dimensions defined in terms of evaluations and quadratic norms of multivariable little $q$-Jacobi polynomials, which interpolate between the various classical dimensions. The generalized dimensions themselves are shown to have representation theoretic interpretations as the quantum dimensions of irreducible spherical quantum representations associated to quantum complex Grassmannians.Comment: 41 pages, final version to appear in IMR

On discrete q-ultraspherical polynomials and their duals

We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q), which corresponds to a=b=-c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a< q^{-1}). Since P_n(x;q^\alpha, q^\alpha, -q^\alpha; q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q->1, this family represents yet another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. The dual family with respect to the polynomials P_n(x;a,a,-a;q) (i.e., the dual discrete q-ultraspherical polynomials) corresponds to the indeterminate moment problem, that is, these polynomials have infinitely many orthogonality relations. We find orthogonality relations for these polynomials, which have not been considered before. In particular, extremal orthogonality measures for these polynomials are derived.Comment: 14 page