Direct spreading measures of Laguerre polynomials

✍ Scribed by P. Sánchez-Moreno; D. Manzano; J.S. Dehesa

• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 308 KB
• Volume: 235
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2010.07.022

✦ Synopsis

The direct spreading measures of the Laguerre polynomials L (α) n (x), which quantify the distribution of its Rakhmanov probability density ρ n,α (x) = 1

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, α) by means of two error-free computing approaches; one makes use of the Lauricella function

, 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).