Algebraic Lq-norms and complexity-like properties of Jacobi polynomials: Degree and parameter asymptotics

Abstract

The Jacobi polynomials P<^>oa,ss THORN n oxTHORN conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two-parameter weight function o1 xTHORNao1thornxTHORN ss, a,ss > 1, on the interval 1/21, thorn1. The spreading of its associated probability density (i.e., the Rakhmanov density) over the support interval has been quantified, beyond the dispersion measures (moments around the origin, variance), by the algebraic Lq-norms (Shannon and Renyi entropies) and the monotonic complexity-like measures of Cramer-Rao, Fisher-Shannon, and LMC (LopezRuiz, 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 oa, ss THORN are large. In this work, we determine in a simple, compact form the leading term of the entropic and complexity-like properties of the Jacobi polynomials in the two extreme situations: (n!8; fixed a, ss) and (a!8; fixed n, ss). These two asymptotics are relevant per se and because they control the physical entropy and complexity measures of the high energy (Rydberg) and high dimensional (pseudoclassical) states of many exactly, conditional exactly, and quasi-exactly solvable quantum- mechanical potentials which model numerous atomic and molecular systems.