Direct spreading measures of Laguerre polynomials

• Published in: Journal of computational and applied mathematics Vol. 235; no. 5; pp. 1129 - 1140
• Main Authors: Sánchez-Moreno, P., Manzano, D., Dehesa, J.S.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 2011; Elsevier
• Subjects: Asymptotic properties; Bell polynomials; Calculus of variations and optimal control; Combinatorial analysis; Combinatorics; Combinatorics. Ordered structures; Computation; Computation of information measures; Density; Exact sciences and technology; Fisher information; Laguerre polynomials; Lauricella functions; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Orthogonal polynomials; Renyi entropy; Sciences and techniques of general use; Shannon entropy; Special functions; Spreading; Spreading lengths; Standard deviation; 33C45; Computation of information measures; Lauricella functions; 65C60; Fisher information; Laguerre polynomials; 62B10; Bell polynomials; 94A17; Orthogonal polynomials; Shannon entropy; Spreading lengths; Renyi entropy; Error estimation; Optimization method; Entropy; Probability distribution; Variational calculus; Combinatorics; Linearization; Mathematical programming; Laguerre polynomial; Orthogonal polynomial; Bell polynomial; Numerical analysis; Applied mathematics; Information theory
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2010.07.022

The direct spreading measures of the Laguerre polynomials L n ( α ) ( x ) , which quantify the distribution of its Rakhmanov probability density ρ n , α ( x ) = 1 d n 2 x α e − x [ L n ( α ) ( 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 2 q ∈ N ) is also found in terms of ( n , α ) by means of two error-free computing approaches; one makes use of the Lauricella function F A ( 2 q + 1 ) ( 1 q , … , 1 q ; 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 ).