Algebraic $\mathcal{L}_{q}$-norms and complexity-like properties of Jacobi polynomials-Degree and parameter asymptotics

TL;DR

This paper derives simple formulas for the entropy and complexity measures of Jacobi polynomials in two key asymptotic regimes, aiding understanding of high-energy quantum states.

Contribution

It provides compact, analytical expressions for entropy and complexity measures of Jacobi polynomials in the limits of large degree and large parameters, which were previously complex to compute.

Findings

Explicit asymptotic formulas for entropic measures

Simplified expressions for complexity-like quantities

Relevance to high-energy and high-dimensional quantum states

Abstract

The Jacobi polynomials $\hat{P}_n^{(\alpha,\beta)}(x)$ conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function $(1-x)^\alpha (1+x)^\beta, \alpha,\beta>-1,$ on the interval $[-1,+1]$. The spreading of its associated probability density (i.e., the Rakhmanov density) over the orthogonality support has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic $\mathfrak{L}_{q}$-norms (Shannon and R\'enyi entropies) and the monotonic complexity-like measures of Cram\'er-Rao, Fisher-Shannon and LMC (L\'opez-Ruiz, Mancini and Calbet) types. These quantities, however, have been often determined in an analytically highbrow, non-handy way; specially when the degree or the parameters $(\alpha,\beta)$ are large. In this work, we determine in a simple, compact form the entropic and complexity-like properties of the Jacobi polynomials in the two extremal situations: ($n\rightarrow \infty$; fixed $\alpha,\beta$) and ($\alpha\rightarrow \infty$; fixed $n,\beta$). These two asymptotics are relevant \textit{per se} and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of numerous supersymmetric quantum-mechanical systems.