Direct spreading measures of Laguerre polynomials

  • Authors:

  • P. Sánchez-Moreno;D. Manzano;J. S. Dehesa

  • Affiliations:

  • Departamento de Matemática Aplicada, Universidad de Granada, Granada, Spain and Instituto "Carlos I" de Física Teórica y Computacional, Universidad de Granada, Granada, Spain;Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada, Spain and Instituto "Carlos I" de Física Teórica y Computacional, Universidad de Granada, ...;Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, Granada, Spain and Instituto "Carlos I" de Física Teórica y Computacional, Universidad de Granada, ...

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2011

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Abstract

The direct spreading measures of the Laguerre polynomials L"n^(^@a^)(x), which quantify the distribution of its Rakhmanov probability density @r"n","@a(x)=1d"n^2x^@ae^-^x[L"n^(^@a^)(x)]^2 along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar root-mean square or standard deviation and the information-theoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order q (such that 2q@?N) is also found in terms of (n,@a) by means of two error-free computing approaches; one makes use of the Lauricella function F"A^(^2^q^+^1^)(1q,...,1q;1), which is based on the Srivastava-Niukkanen linearization relation of Laguerre polynomials, and another one utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by the use of an information-theoretic optimization procedure. Finally, all these spreading measures are mutually compared and computationally analyzed; in particular, it is found that the apparent quasilinear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for n@?1).