  The density matrix for the impenetrable Bose gas in Dirichlet and Neumann
boundary conditions can be written in terms of $<\prod_{l=1}^n|
\cos\phi_1-\cos\theta_l|
  |\cos\phi_2-\cos\theta_l|>$, where the average is with respect to the
eigenvalue probability density function for random unitary matrices from the
classical groups $Sp(n)$ and $O^+(2n)$ respectively. In the large $n$ limit
log-gas considerations imply that the average factorizes into the product of
averages of the form $<\prod_{l=1}^n|\cos\phi-\cos\theta_l>$. By changing
variables this average in turn is a special case of the function of $t$
obtained by averaging $\prod_{l=1}^n| t-x_l|^{2q}$ over the Jacobi unitary
ensemble from random matrix theory. The latter task is accomplished by a
duality formula from the theory of Selberg correlation integrals, and the large
$n$ asymptotic form is obtained. The corresponding large $n$ asymptotic form of
the density matrix is used, via the exact solution of a particular integral
equation, to compute the asymptotic form of the low lying effective single
particle states and their occupations, which are proportional to $\sqrt{N}$.
