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

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 Rényi entropies) and the monotonic complexity-like measures of Cramér-Rao, Fisher-Shannon and LMC (López-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: ($n\to \infty$; fixed $\alpha ,\beta$) and ($\alpha \to \infty$; 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.

11 pages, 1 figure

References in corpus (1)

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