Direct spreading measures of Laguerre polynomials
Abstract
The direct spreading measures of the Laguerre polynomials, which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar rootmeansquare or standard deviation and the informationtheoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order q (such that 2q is a natural number) is also found in terms of the polynomials parameters by means of two errorfree computing approaches; one makes use of the Lauricella functions, which is based on the SrivastavaNiukkanen linearization relation of Laguerre polynomials, and another one which utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmicfunctional form, but its asymptotics is provided and sharp bounds are obtained by use of an informationtheoretic 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).
 Publication:

arXiv eprints
 Pub Date:
 September 2010
 arXiv:
 arXiv:1009.0289
 Bibcode:
 2010arXiv1009.0289S
 Keywords:

 Mathematical Physics;
 Computer Science  Information Theory;
 Quantum Physics
 EPrint:
 15 pages, 5 figures, accepted in Journal of Computational and Applied Mathematics