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

• Published in: International journal of quantum chemistry Vol. 122; no. 6
• Main Authors: Sobrino, Nahual, S.‐Dehesa, Jesus
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15.03.2022; Wiley Subscription Services, Inc
• Subjects: Algebra; Asymptotic properties; Chemistry; Complexity; Density; Norms; Parameters; Physical chemistry; Polynomials; Quantum physics; Weighting functions
• ISSN: 0020-7608; 1097-461X
• DOI: 10.1002/qua.26858

The Jacobi polynomials P^nαβx conform the canonical family of hypergeometric orthogonal polynomials (HOPs) with the two‐parameter weight function 1−xα1+xβ,α,β>−1, on the interval −1+1. 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 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 αβ 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→∞; fixed α,β) and (α→∞; fixed n,β). 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. The Cramér–Rao, Fisher–Shannon and LMC complexity‐like measures of the Jacobi polynomials are determined when the polynomial's degree and parameter tend to infinity. They are given by the leading term of the degree and parameter asymptotics of the corresponding statistical properties of the associated probability density. These results are not only interesting per se, but also because they control the physical complexities of some atoms and molecules at the highly excited Rydberg and pseudoclassical states.