zbMATH — the first resource for mathematics

Complexity analysis of hypergeometric orthogonal polynomials. (English) Zbl 1351.60036
Summary: The complexity measures of the Crámer-Rao, Fisher-Shannon and LMC (López-Ruiz, Mancini and Calvet) types of the Rakhmanov probability density \(\rho_n(x) = \omega(x) p_n^2(x)\) of the polynomials \(p_n(x)\) orthogonal with respect to the weight function \(\omega(x)\), \(x \in(a, b)\), are used to quantify various two-fold facets of the spreading of the Hermite, Laguerre and Jacobi systems all over their corresponding orthogonality intervals in both analytical and computational ways. Their explicit (Crámer-Rao) and asymptotical (Fisher-Shannon, LMC) values are given for the three systems of orthogonal polynomials. Then, these complexity-type mathematical quantities are numerically examined in terms of the polynomial’s degree \(n\) and the parameters which characterize the weight function. Finally, several open problems about the generalized hypergeometric functions of Lauricella and Srivastava-Daoust types, as well as on the asymptotics of weighted \(L_q\)-norms of Laguerre and Jacobi polynomials are pointed out.
60F99 Limit theorems in probability theory
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Full Text: DOI arXiv
[1] Comission, E., Leonhard euler’s opera omnia, (2003), Birkhäuser Verlag
[2] Badii, R.; Politi, A., Complexity: hierarchical structure and scaling in physics, (1997), Henry Holt, New York · Zbl 1042.82500
[3] Gell-Mann, M.; Lloyd, S., Complexity, 2, 44, (1996) · Zbl 1294.94011
[4] Holland, J., Signals and boundaries: building blocks for complex adaptive systems, (2012), M.I.T. Press, Cambridge, MA
[5] Nikiforov, A. F.; Uvarov, V. B., Special functions in mathematical physics, (1988), Birkäuser-Verlag, Basel · Zbl 0645.33019
[6] Temme, N. M., Special functions: an introduction to the classical functions of mathematical physics, (1996), Wiley-Intersciente, New York · Zbl 0863.33002
[7] Galindo, A.; Pascual, P., Quantum mechanics, (1987), Cambridge University Press, Cambridge · Zbl 0824.00008
[8] Goldreich, O., Computational complexity: A conceptual perspective, (2008), Cambridge University Press, Cambridge · Zbl 1154.68056
[9] Cubitt, T., Advanced quantum information theory, (2014), Cambridge University Press, Cambridge
[10] Angulo, J. C.; Antolín, J.; Esquivel, R. O., Atomic and molecular complexities: their physical and chemical interpretations, (Sen, K. D., Statistical Complexities: Applications in Electronic Structure, (2011), Springer, Berlin)
[11] Dehesa, J. S.; Lopez-Rosa, S.; Manzano, D., Entropy and complexity analyses of \(d\)-dimensional quantum systems, (Sen, K. D., Statistical Complexities: Applications in Electronic Structure, (2011), Springer, Berlin)
[12] Martin, M.; Plastino, A.; Rosso, O., J. Math. Chem., 369, 439-462, (2006)
[13] Dehesa, J. S.; Lopez-Rosa, S.; Manzano, D., Configuration complexities of hydrogenic systems, Eur. Phys. J. D, 55, 539-548, (2009)
[14] Lopez-Rosa, S.; Manzano, D.; Dehesa, J., Complexity of d-dimensional hydrogenic systems in position and momentum spaces, Physica A, 388, 3273-3281, (2009)
[15] Molina-Espíritu, M.; Esquivel, R. O.; Angulo, J.; Antolín, J.; Dehesa, J. S., Information-theoretical complexity for the hydrogenic identity \(S_N 2\) exchange reaction, J. Math. Chem., 50, 1882-1900, (2012) · Zbl 1319.92062
[16] López-Rosa, S.; Toranzo, I. V.; Sánchez-Moreno, P.; Dehesa, J. S., Entropy and complexity analysis of hydrogenic Rydberg atoms, J. Math. Phys., 54, 052109, (2013) · Zbl 1282.81054
[17] Rakhmanov, E., On the asymptotics of the ratio of orthogonals polynomials, Math. USSR, 32, 199-213, (1977) · Zbl 0401.30033
[18] Rényi, A., Probability theory, (1970), North Holland, Amsterdam · Zbl 0206.18002
[19] Shannon, C. E.; Weaver, W., The mathematical theory of communication, (1949), University of Illinois Press, Urbana · Zbl 0041.25804
[20] Frieden, B. R., Science from Fisher information, (2004), Cambridge University Press, Cambridge · Zbl 0881.60016
[21] Fisher, R. A., Theory of statistical estimation, Proc. Camb. Phil. Soc., 22, 700-725, (1925), Reprinted in Collected Papers of R.A. Fisher, edited by J.H. Bennet (University of Adelaide Press, South Australia), 1972, 15-40 · JFM 51.0385.01
[22] J. W. Hall, M., Universal geometric approach to uncertainty, entropy and information, Phys. Rev. A, 59, 2602-2615, (1999)
[23] Dembo, A.; Cover, T. M.; Thomas, J. A., Information theoretic inequalities, IEEE Trans. Inform. Theory, 37, 1501-1528, (1991) · Zbl 0741.94001
[24] Dehesa, J. S.; Sánchez-Moreno, P.; Yáñez, R. J., Cramér-Rao information plane of orthogonal hypergeometric polynomials, J. Comput. Appl. Math., 186, 523-541, (2006) · Zbl 1086.33010
[25] Antolín, J.; Angulo, J. C., Complexity analysis of ionization processes and isoelectronic series, Int. J. Quant. Chem., 109, 586-593, (2009)
[26] Angulo, J. C.; Antolín, J.; Sen, K. D., Fisher-Shannon plane and statistical complexity of atoms, Phys. Lett. A, 372, 670, (2008) · Zbl 1217.81155
[27] Romera, E.; Dehesa, J., The Fisher-Shannon information plane, an electron correlation tool, J. Chem. Phys., 120, 8906-8912, (2004)
[28] Catalan, R. G.; Garay, J.; López-Ruiz, R., Features of the extension of a statistical measure of complexity to continuous systems, Phys. Rev. E, 66, 011102, (2002)
[29] Yamano, T., A statistical complexity measure with nonextensive entropy and quasi-multiplicativity, J. Math. Phys., 45, 1974-1987, (2004) · Zbl 1071.94005
[30] Yamano, T., A statistical measure of complexity with nonextensive entropy, Physica A, 340, 131-137, (2004)
[31] Sánchez-Moreno, P.; Manzano, D.; Dehesa, J. S., Direct spreading measures of Laguerre polynomials, J. Comput. Appl. Math., 235, 1129-1140, (2011) · Zbl 1223.33017
[32] Sanchez-Ruiz, J.; Dehesa, J. S., Fisher information of orthogonal hypergeometric polynomials, J. Comput. Appl. Math., 182, 150-164, (2005) · Zbl 1081.33017
[33] Sánchez-Moreno, P.; Dehesa, J. S.; Manzano, D.; Yáñez, R. J., Spreading lengths of Hermite polynomials, J. Comput. Appl. Math., 233, 2136-2148, (2010) · Zbl 1188.33017
[34] Aptekarev, A. I.; Dehesa, J. S.; Sánchez-Moreno, P.; Tulyakov, D. N., Asymptotics of \(L_p\)-norms of Hermite polynomials and Rényi entropy of Rydberg oscillator states, Contemp. Math., 578, 19-29, (2012) · Zbl 1318.94027
[35] Olver, F. W.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., Handbook of mathematical functions, (2010), Cambridge University Press, Cambridge · Zbl 1198.00002
[36] Srivastava, H. M.; Karlsson, P. W., Multiple Gaussian hypergeometric series, (1985), John Wiley and Sons, New York · Zbl 0552.33001
[37] Guerrero, A.; Sánchez-Moreno, P.; Dehesa, J. S., Information-theoretic lengths of Jacobi polynomials, J. Phys. A: Math. Theor., 43, 305203, (2010) · Zbl 1220.33009
[38] Sánchez-Moreno, P.; Zarzo, A.; Dehesa, J. S., Rényi entropies, \(L_q\) norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions, Appl. Math. Comput., 223, 25-33, (2013) · Zbl 1329.33013
[39] Dehesa, J. S.; Guerrero, A.; López, J. L.; Sánchez-Moreno, P., Asymptotics (\(p\) to infinity) of \(L_p\)-norms of hypergeometric orthogonal polynomials, J. Math. Chem., 52, 283-300, (2014) · Zbl 1286.81060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.