Orthogonal Polynomials from Hermitian Matrices II

TL;DR

This paper extends the theory of classical orthogonal polynomials derived from hermitian matrices, including the big q-Jacobi family, by recovering self-adjointness in an extended space and constructing complete eigenvector sets.

Contribution

It introduces a method to recover self-adjointness for unbounded Jacobi matrices and constructs complete eigenvector bases involving q-Meixner and q-Charlier polynomials.

Findings

Jackson integral measures for big q-Jacobi polynomials derived from self-adjointness recovery.

Dual q-Meixner polynomials are used to complete the eigenvector basis.

An alternative method using the closure relation is applied to various polynomial families.

Abstract

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.