  The spreading properties of the stationary states of the quantum
multidimensional harmonic oscillator are analytically discussed by means of the
main dispersion measures (radial expectation values) and the fundamental
entropy-like quantities (Fisher information, Shannon and R\'enyi entropies,
disequilibrium) of its quantum probability distribution together with their
associated uncertainty relations. They are explicitly given, at times in a
closed compact form, by means of the potential parameters (oscillator strength,
dimensionality, $D$) and the hyperquantum numbers
$(n_r,\mu_1,\mu_2,\ldots,\mu_{D-1})$ which characterize the state. Emphasis is
placed on the highly-excited Rydberg (high radial hyperquantum number $n_r$,
fixed $D$) and the high-dimensional (high $D$, fixed hyperquantum numbers)
states. We have used a methodology where the theoretical determination of the
integral functionals of the Laguerre and Gegenbauer polynomials, which describe
the spreading quantities, leans heavily on the algebraic properties and
asymptotical behavior of some weighted $\mathfrak{L}_{q}$-norms of these
orthogonal functions.
