Entropy-Like Properties and <inline-formula><math display="inline"><semantics><msub><mi mathvariant="fraktur">L</mi><mi>q</mi></msub></semantics></math></inline-formula>-Norms of Hypergeometric Orthogonal Polynomials: Degree Asymptotics

• Main Author: Jesús S. Dehesa
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Symmetry
• Subjects: special functions; classical orthogonal polynomials; entropy-like measures; asymptotics
• Online Access: https://www.mdpi.com/2073-8994/13/8/1416

In this work, the spread of hypergeometric orthogonal polynomials (HOPs) along their orthogonality interval is examined by means of the main entropy-like measures of their associated Rakhmanov’s probability density—so, far beyond the standard deviation and its generalizations, the ordinary moments. The Fisher information, the Rényi and Shannon entropies, and their corresponding spreading lengths are analytically expressed in terms of the degree and the parameter(s) of the orthogonality weight function. These entropic quantities are closely related to the gradient functional (Fisher) and the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">L</mi><mi>q</mi></msub></semantics></math></inline-formula>-norms (Rényi, Shannon) of the polynomials. In addition, the degree asymptotics for these entropy-like functionals of the three canonical families of HPOs (i.e., Hermite, Laguerre, and Jacobi polynomials) are given and briefly discussed. Finally, a number of open related issues are identified whose solutions are both physico-mathematically and computationally relevant.