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The three canonical families of the hypergeometric orthogonal polynomials(Hermite, Laguerre and Jacobi) control the physical wavefunctions of the boundstationary states of a great deal of quantum systems. The algebraic$\mathfrak{L}_{q}$-norms of these polynomials describe many physical, chemicaland information-theoretical properties of these systems, such as e.g. thekinetic and Weizs\"acker energies, the position and momentum expectationvalues, the R\'enyi and Shannon entropies and the Cram\'er-Rao, theFisher-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 theunweighted and weighted $\mathfrak{L}_{q}$-norms for these orthogonalpolynomials. This study has been motivated by the application of thesealgebraic norms to the energetic, entropic and complexity-like properties ofthe highly-excited Rydberg and high-dimensional pseudo-classical states ofharmonic (oscillator-like) and Coulomb (hydrogenic) systems, and other quantumsystems subject to central potentials of anharmonic type.

Parameter and $q$ asymptotics of $\mathfrak{L}_{q}$-norms of hypergeometric orthogonal polynomials