Spreading lengths of Hermite polynomials

Abstract

The Renyi, Shannon and Fisher spreading lengths of the classical or hypergeometric orthogonal polynomials, which are quantifiers of their distribution all over the orthogonality interval, are defined and investigated. These information-theoretic measures of the associated Rakhmanov probability density, which are direct measures of the polynomial spreading in the sense of having the same units as the variable, share interesting properties: invariance under translations and reflections, linear scaling and vanishing in the limit that the variable tends towards a given definite value. The expressions of the Renyi and Fisher lengths for the Hermite polynomials are computed in terms of the polynomial degree. The combinatorial multivariable Bell polynomials, which are shown to characterize the finite power of an arbitrary polynomial, play a relevant role for the computation of these information-theoretic lengths. Indeed these polynomials allow us to design an error-free computing approach for the entropic moments (weighted L^q-norms) of Hermite polynomials and subsequently for the Renyi and Tsallis entropies, as well as for the Renyi spreading lengths. Sharp bounds for the Shannon length of these polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, it is computationally proved the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation. Finally, the application to the most popular quantum-mechanical prototype system, the harmonic oscillator, is discussed and some relevant asymptotical open issues related to the entropic moments mentioned previously are posed.

16 pages, 4 figures. Journal of Computational and Applied Mathematics (2009), doi:10.1016/j.cam.2009.09.043

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• A Statistical Measure of Complexity
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• Schrodinger equation from an exact uncertainty principle
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Cited by in corpus (9)

• Jensen divergence based on Fisher's information
• Entropic measures of Rydberg-like harmonic states
• Direct spreading measures of Laguerre polynomials
• Information Entropy of conditionally exactly solvable potentials
• Information-theoretic-based spreading measures of orthogonal polynomials
• Linearization and Krein-like functionals of hypergeometric orthogonal polynomials
• Complexity analysis of hypergeometric orthogonal polynomials
• Evolution of capacity of entanglement and modular entropy in harmonic chains and scalar fields
• Complexity-like properties and parameter asymptotics of ${\mathfrak{L}}_q$-norms of Laguerre and Gegenbauer polynomials