Parameter and q asymptotics of Lq‐norms of hypergeometric orthogonal polynomials

• Published in: International journal of quantum chemistry Vol. 123; no. 2
• Main Authors: Sobrino, Nahual, Dehesa, Jesus S.
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 15.01.2023; Wiley Subscription Services, Inc
• Subjects: Algebra; Anharmonicity; Asymptotic properties; Chemistry; Complexity; Continuity (mathematics); Mathematical analysis; Norms; Oscillators; Parameters; Physical chemistry; Polynomials; Quantum physics; Real variables; Wave functions; Weighting functions
• ISSN: 0020-7608; 1097-461X
• DOI: 10.1002/qua.27013

The three canonical families of the hypergeometric orthogonal polynomials in a continuous real variable (Hermite, Laguerre, and Jacobi) control the physical wavefunctions of the bound stationary states of a great number of quantum systems [Correction added after first online publication on 21 December, 2022. The sentence has been modified.]. The algebraic Lq‐norms of these polynomials describe many chemical, physical, and information theoretical properties of these systems, such as, for example, the kinetic and Weizsäcker energies, the position and momentum expectation values, the Rényi and Shannon entropies and the Cramér‐Rao, the Fisher‐Shannon and LMC measures of complexity. In this work, we examine review and solve the q‐asymptotics and the parameter asymptotics (i.e., when the weight function's parameter tends towards infinity) of the unweighted and weighted Lq‐norms for these orthogonal polynomials. This study has been motivated by the application of these algebraic norms to the energetic, entropic, and complexity‐like properties of the highly excited Rydberg and high‐dimensional pseudo‐classical states of harmonic (oscillator‐like) and Coulomb (hydrogenic) systems, and other quantum systems subject to central potentials of anharmonic type (such as, e.g., some molecular systems) [Correction added after first online publication on 21 December, 2022. Oscillatorlike has been changed to oscillator‐like.]. The weighted Lq‐norms of orthogonal polynomials are determined when q and the polynomial's parameter tend to infinity. They are given in this work by the leading term of the q and parameter asymptotics of the corresponding quantities of the associated probability density. These results are not only interesting per se, but also because they control many chemical and entropic properties of some atoms and molecules at the highly excited Rydberg and quasiclassical states.