Abstract
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.
Highlights
Hypergeometric orthogonal polynomials (HOPs), called Shohat–Favard polynomials and classical orthogonal polynomials, play a key role in the development of the theory of special functions, and they are instrumental in numerous scientific problems, ranging from approximation theory to quantum theory and mathematical physics [1,2,3,4,5,6,7,8]
In order for all these quantifiers to be mutually compared on the same footing, we have examined the corresponding information-theoretic lengths of Fisher, Shannon and Rényi types [22,23,24,25]
We describe the spreading measures of dispersion and information-theoretic (Fisher information, Shannon entropy, Rényi entropies) types for a random variable X characterized by the continuous probability density ρ( x ), x ∈ Λ ⊆ R
Summary
Hypergeometric orthogonal polynomials (HOPs), called Shohat–Favard polynomials and classical orthogonal polynomials, play a key role in the development of the theory of special functions, and they are instrumental in numerous scientific problems, ranging from approximation theory to quantum theory and mathematical physics [1,2,3,4,5,6,7,8]. In order for all these quantifiers to be mutually compared on the same footing, we have examined the corresponding information-theoretic lengths of Fisher, Shannon and Rényi types [22,23,24,25] These quantities, together with the standard deviation, are direct spreading measures [26]. We update the analytical determination of the spreading measures of the HOPs with emphasis on the entropy-like quantities and Lq -norms because of their relevance in the information theory of special functions and quantum systems, and to facilitate its numerical and symbolic computation. Some concluding remarks are pointed out and a number of open related issues are identified
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