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

TL;DR

This paper analyzes the asymptotic behavior of $rak{L}_q$-norms of Hermite, Laguerre, and Jacobi polynomials, revealing their implications for quantum systems' energetic and informational properties.

Contribution

It provides a detailed study of the $q$- and parameter asymptotics of $rak{L}_q$-norms for hypergeometric orthogonal polynomials, with applications to quantum physics.

Findings

Derived asymptotic formulas for $rak{L}_q$-norms

Linked norms to quantum energetic and entropic measures

Applicable to high-energy quantum states and systems

Abstract

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.