Parameter and q asymptotics of Lq-norms of hypergeometric orthogonal polynomials

Abstract

The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic Lq-norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér-Rao, the Fisher-Shannon and LMC measures of complexity. In this work, we examine 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 Lq-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 (such as, e.g., some molecu- lar systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator-like.].