Complexity-like properties and parameter asymptotics of $\mathfrak{L}_{q}$-norms of Laguerre and Gegenbauer polynomials

TL;DR

This paper investigates the complexity measures of Rakhmanov's probability densities linked to Laguerre and Gegenbauer polynomials, analyzing their asymptotic behavior in parameters and degrees, with implications for quantum systems in high-energy and high-dimensional states.

Contribution

It provides analytical expressions and asymptotic analysis of complexity measures for Laguerre and Gegenbauer polynomials, extending understanding beyond the Cramér-Rao measure.

Findings

Analytical formulas for Fisher-Shannon and LMC complexities in terms of polynomial degree and parameters.

Asymptotic behavior of these complexity measures as parameters tend to infinity.

Implications for charge and momentum distributions in high-energy and high-dimensional quantum states.

Abstract

The main monotonic statistical complexity-like measures of the Rakhmanov's probability density associated to the hypergeometric orthogonal polynomials (HOPs) in a real continuous variable, each of them quantifying two configurational facets of spreading, are examined in this work beyond the Cram\'er-Rao one. The Fisher-Shannon and LMC (L\'opez-Ruiz-Mancini-Calvet) complexity measures, which have two entropic components, are analytically expressed in terms of the degree and the orthogonality weight's parameter(s) of the polynomials. The degree and parameter asymptotics of these two-fold spreading measures are shown for the parameter-dependent families of HOPs of Laguerre and Gegenbauer types. This is done by using the asymptotics of the R\'enyi and Shannon entropies, which are closely connected to the $\mathfrak{L}_{q}$-norms of these polynomials, when the weight's parameter tends towards infinity. The degree and parameter asymptotics of these Laguerre and Gegenbauer algebraic norms control the radial and angular charge and momentum distributions of numerous relevant multidimensional physical systems with a spherically-symmetric quantum-mechanical potential in the high-energy (Rydberg) and high-dimensional (quasi-classical) states, respectively. This is because the corresponding states' wavefunctions are expressed by means of the Laguerre and Gegenbauer polynomials in both position and momentum spaces.