  The three canonical families of the hypergeometric orthogonal polynomials
(Hermite, Laguerre and Jacobi) control the physical wavefunctions of the bound
stationary states of a great deal of quantum systems. The algebraic
$\mathfrak{L}_{q}$-norms of these polynomials describe many physical, chemical
and information-theoretical properties of these systems, such as e.g. the
kinetic and Weizs\"acker energies, the position and momentum expectation
values, the R\'enyi and Shannon entropies and the Cram\'er-Rao, the
Fisher-Shannon and LMC measures of complexity. In this work we examine,
partially review and solve the $q$-asymptotics and the parameter asymptotics
(i.e., when the weight function's parameter tends towards infinity) of the
unweighted and weighted $\mathfrak{L}_{q}$-norms for these orthogonal
polynomials. This study has been motivated by the application of these
algebraic norms to the energetic, entropic and complexity-like properties of
the highly-excited Rydberg and high-dimensional pseudo-classical states of
harmonic (oscillator-like) and Coulomb (hydrogenic) systems, and other quantum
systems subject to central potentials of anharmonic type.
