Complexity-like properties and parameter asymptotics of -norms of Laguerre and Gegenbauer polynomials

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 Cramér–Rao one. The Fisher–Shannon and LMC López-Ruiz–Mancini–Calvet (LMC) complexity measures, which have two entropic components, are analytically expressed in terms of the degree and the orthogonality weight-function’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 Rényi and Shannon entropies, which are closely connected to the -norms of these polynomials, when the weight-function’s parameter tends toward 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.