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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.

MSC:
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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References:
[1] Temme, N.M., Special functions: an introduction to the classical functions of mathematical physics, (1996), Wiley-Intersciente · Zbl 0863.33002
[2] Nikiforov, A.F.; Uvarov, V.B., Special functions in mathematical physics, (1988), Birkäuser-Verlag Basel · Zbl 0645.33019
[3] Rakhmanov, E.A., On the asymptotics of the ratio of orthogonal polynomials, Math. USSR sb., 32, 199-213, (1977) · Zbl 0401.30033
[4] Fisher, R.A., Theory of statistical estimation, (), 700-725, reprinted in Collected Papers of R.A. Fisher 1972, pp. 15-40 · JFM 51.0385.01
[5] Renyi, A., Probability theory, (1970), North Holland Amsterdam · Zbl 0206.18002
[6] Shannon, C.E.; Weaver, W., The mathematical theory of communication, (1949), University of Illinois Press Urbana · Zbl 0041.25804
[7] Dehesa, J.S.; Martínez-Finkelshtein, A.; Sánchez-Ruiz, J., Quantum information entropies and orthogonal polynomials, J. comput. appl. math., 133, 23-46, (2001) · Zbl 1008.81014
[8] Aptekarev, A.I.; Martínez-Finkelshtein, A.; Dehesa, J.S., Asymptotics of orthogonal polynomials entropy, J. comput. appl. math., (2009) · Zbl 1177.33018
[9] Aptekarev, A.I.; Buyarov, V.S.; Dehesa, J.S., Asymptotic behavior of the \(L^p\)-norms and the entropy for general orthogonal polynomials, Russian acad. sci. sb. math., 82, 373-395, (1995)
[10] Dehesa, J.S.; van Assche, W.; Yáñez, R.J., Position and momentum information entropies of the D-dimensional harmonic oscillator and hydrogen atom, Phys. rev. A, 50, 3065-3079, (1994)
[11] Dehesa, J.S.; van Assche, W.; Yáñez, R.J., Information entropy of classical orthogonal polynomials and their application to the harmonic oscillator and Coulomb potentials, Methods appl. anal., 4, 91-110, (1997) · Zbl 0877.33005
[12] Buyarov, V.S.; López-Artés, P.; Martínez-Finkelshtein, A.; van Assche, W., Information entropy of Gegenbauer polynomials, J. phys. A: math. gen., 33, 6549-6560, (2000) · Zbl 1008.81015
[13] de Vicente, J.I.; Gandy, S.; Sánchez-Ruiz, J., Information entropy of Gegenbauer polynomials, J. phys. A: math. theor., 40, 8345-8361, (2007) · Zbl 1120.33011
[14] Sánchez-Ruiz, J.; Dehesa, J.S., Fisher information of orthogonal hypergeometric polynomials, J. comput. appl. math., 182, 150-164, (2005) · Zbl 1081.33017
[15] Dehesa, J.S.; Sánchez-Moreno, P.; Yáñez, R.J., Cramer – rao information plane of orthogonal hypergeometric polynomials, J. comput. appl. math., 186, 523-541, (2006) · Zbl 1086.33010
[16] Hall, M.J.W., Universal geometric approach to uncertainty, entropy and information, Phys. rev. A, 59, 2602-2615, (1999)
[17] Shenton, L.R., Efficiency of the method of moments and the gram – charlier type A distribution, Biometrika, 38, 58-73, (1951) · Zbl 0042.38503
[18] Sichel, H.S., The metod of moments and its applications to type VII populations, Biometrika, 36, 404, (1949)
[19] Sichel, H.S., Fitting growth and frequency curves by the method of frequency moments, J. roy. statist. soc., A110, 337-347, (1947) · Zbl 0031.06101
[20] Comtet, L., Advanced combinatorics, (1974), D. Reidel Publ Dordrecht
[21] Godsil, C.D., Combinatorics, 1, 251-262, (1982)
[22] Kendall, M.G.; Stuart, A., The advanced theory of statistics, vol. 1, (1969), Charles Griffin Co. London · Zbl 0223.62001
[23] Yule, G.U., On some properties of the normal distribution, univariate and bivariate, based on the sum of squares of frequencies, Biometrika, 30, 1-10, (1938) · Zbl 0019.12902
[24] Romera, E.; Angulo, J.C.; Dehesa, J.S., Reconstruction of a density from its entropic moments, (), p. 449 · Zbl 1009.44003
[25] Tsallis, C., Possible generalization of boltzmann – gibbs statistics, J. stat. phys., 52, 479-487, (1998) · Zbl 1082.82501
[26] Brukner, C.; Zeilinger, A., Conceptual inadequacy of the Shannon information in quamtum mechanics, Phys. rev. A, 63, 022113, (2001)
[27] Zurek, W.H.; Habid, S.; Paz, J.P., Coherent states via decoherence, Phys. rev. lett., 70, 1187-1190, (1993)
[28] Onicescu, O., Theorie de l’information. energie informationelle, C.R. acad. sci. Paris A, 263, 25, (1966)
[29] Heller, E., Quantum localization and the rate of exploration of phase space, Phys. rev. A, 35, 1360-1370, (1987)
[30] Anteneodo, C.; Plastino, A.R., Some features of the López – ruiz – mancini – calbet (LMC) statistical measure of complexity, Phys. lett. A, 223, 348-354, (1996) · Zbl 1037.82500
[31] López-Ruiz, R.; Mancini, H.L.; Calbet, X., A statistical measure of complexity, Phys. lett. A, 209, 321-326, (1995)
[32] Zyckowski, K., Indicators of quantum chaos based on eigenvector statistics, J. phys. A: math. gen., 23, 4427-4438, (1990)
[33] Mirbach, B.; Korch, J.H., A generalized entropy measuring quantum localization, Ann. phys., 265, 80-97, (1998) · Zbl 0946.81013
[34] Hall, M.J.W., Exact uncertainty relations, Phys. rev. A, 64, 052103, (2001)
[35] Hall, M.J.W.; Reginatto, M., Schrödinger equation from an exact uncertainty principle, J. phys. A: math. gen., 35, 3289-3303, (2002) · Zbl 1045.81003
[36] Dehesa, J.S.; Martínez-Finkelshtein, A.; Sorokin, V.N., Information-theoretic measures for Morse and Pöschl – teller potentials, Mol. phys., 104, 613-622, (2006)
[37] Dehesa, J.S.; González-Férez, R.; Sánchez-Moreno, P., The Fisher-information-based uncertainty relation, cramer – rao inequality and kinetic energy for the D-dimensional central problem, J. phys. A: math. theor., 40, 1845-1856, (2007) · Zbl 1114.81015
[38] Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I., Integrals and series, (1986), Gordon and Breach Science Publishers Amsterdam · Zbl 0606.33001
[39] Azor, R.; Gillis, J.; Victor, J.D., Combinatorial applications of Hermite polynomials, SIAM J. math. anal., 13, 879-890, (1982) · Zbl 0516.33008
[40] de Sainte-Catherine, M.; Viennot, G., Combinatorial interpretation of integrals of products of Hermite, Laguerre and Tchebycheff polynomials, Lect. notes in math., 1171, 120-128, (1985) · Zbl 0587.05003
[41] Lavoie, J.L., The \(m\)th power of an \(n \times n\) matrix and the Bell polynomials, SIAM J. appl. math., 29, 511-514, (1975) · Zbl 0318.33019
[42] Riordan, J., An introduction to combinatorial analysis, (1980), Princeton Univ. Press New Jersey · Zbl 0436.05001
[43] Aptekarev, A.I.; Buyarov, V.S.; van Assche, W.; Dehesa, J.S., Asymptotics of entropy integrals for orthogonal polynomials, Doklady math., 53, 47-49, (1996) · Zbl 0892.33005
[44] J.I. de Vicente, J. Sánchez-Ruiz, J.S. Dehesa, Information entropy and standard deviation of probability distributions involving orthogonal polynomials. Communication to IWOP 2004, Madrid
[45] Buyarov, V.S.; Dehesa, J.S.; Martínez-Finkelshtein, A.; Sánchez-Lara, J., Computation of the entropy of polynomials orthogonal on an interval, SIAM J. sci. comput., 26, 488-509, (2004) · Zbl 1082.33004
[46] Larsson-Cohn, L., \(L^p\)-norms of Hermite polynomials and an extremal problem on Wiener chaos, Arkiv mat., 40, 133-144, (2002) · Zbl 1021.60043
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