Entropy-Like Properties and Lq-Norms of Hypergeometric Orthogonal Polynomials: Degree Asymptotics

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....

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Bibliographic Details
Published inSymmetry (Basel) Vol. 13; no. 8; p. 1416
Main Author Dehesa, Jesús S.
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 03.08.2021
Subjects
Asymptotic properties
Entropy
Entropy (Information theory)
Fisher information
Hermite polynomials
Information theory
Localization
Norms
Orthogonality
Polynomials
Random variables
Standard deviation
Weighting functions
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Summary: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 Lq-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.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym13081416