  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.
