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Spreading lengths of Hermite polynomials. (English) Zbl 1188.33017
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. The Renyi and Fisher lengths as well as other entropies for the Hermite polynomials are computed in terms of the combinatorial multivariable Bell polynomials. Sharp bounds for the Shannon length of Hermite polynomials are also given by means of an information-theoretic-based optimization procedure. Moreover, the existence of a linear correlation between the Shannon length (as well as the second-order Renyi length) and the standard deviation is computationally proved. 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.

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
94A17 Measures of information, entropy
62B10 Statistical aspects of information-theoretic topics
65C60 Computational problems in statistics (MSC2010)
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