  The R\'enyi entropies $R_{p}[\rho]$, $p>0,\neq 1$ of the highly-excited
quantum states of the $D$-dimensional isotropic harmonic oscillator are
analytically determined by use of the strong asymptotics of the orthogonal
polynomials which control the wavefunctions of these states, the Laguerre
polynomials. This Rydberg energetic region is where the transition from
classical to quantum correspondence takes place. We first realize that these
entropies are closely connected to the entropic moments of the
quantum-mechanical probability $\rho_n(\vec{r})$ density of the Rydberg
wavefunctions $\Psi_{n,l,\{\mu\}}(\vec{r})$; so, to the $\mathcal{L}_{p}$-norms
of the associated Laguerre polynomials. Then, we determine the asymptotics
$n\to\infty$ of these norms by use of modern techniques of approximation theory
based on the strong Laguerre asymptotics. Finally, we determine the dominant
term of the R\'enyi entropies of the Rydberg states explicitly in terms of the
hyperquantum numbers ($n,l$), the parameter order $p$ and the universe
dimensionality $D$ for all possible cases $D\ge 1$. We find that (a) the
R\'enyi entropy power decreases monotonically as the order $p$ is increasing
and (b) the disequilibrium (closely related to the second order R\'enyi
entropy), which quantifies the separation of the electron distribution from
equiprobability, has a quasi-Gaussian behavior in terms of $D$.
