  $N$-dimensional Bessel and Jacobi processes describe interacting particle
systems with $N$ particles and are related to $\beta$-Hermite,
$\beta$-Laguerre, and $\beta$-Jacobi ensembles. For fixed $N$ there exist
associated weak limit theorems (WLTs) in the freezing regime $\beta\to\infty$
in the $\beta$-Hermite and $\beta$-Laguerre case by Dumitriu and Edelman (2005)
with explicit formulas for the covariance matrices $\Sigma_N$ in terms of the
zeros of associated orthogonal polynomials. Recently, the authors derived these
WLTs in a different way and computed $\Sigma_N^{-1}$ with formulas for the
eigenvalues and eigenvectors of $\Sigma_N^{-1}$ and thus of $\Sigma_N$. In the
present paper we use these data and the theory of finite dual orthogonal
polynomials of de Boor and Saff to derive formulas for $\Sigma_N$ from
$\Sigma_N^{-1}$ where, for $\beta$-Hermite and $\beta$-Laguerre ensembles, our
formulas are simpler than those of Dumitriu and Edelman. We use these
polynomials to derive asymptotic results for the soft edge in the freezing
regime for $N\to\infty$ in terms of the Airy function. For $\beta$-Hermite
ensembles, our limit expressions are different from those of Dumitriu and
Edelman.
