Direct spreading measures of Laguerre polynomials

TL;DR

This paper investigates various spreading measures of Laguerre polynomials, including explicit formulas, asymptotic behaviors, and bounds, to understand their probability distribution characteristics along the positive real line.

Contribution

It provides explicit formulas for Fisher and Renyi lengths, asymptotic analysis of Shannon length, and compares these measures to reveal their relationships and behaviors for Laguerre polynomials.

Findings

Fisher length is explicitly derived.

Renyi length is expressed using Lauricella and Bell functions.

Asymptotic behavior of Shannon length is characterized.

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 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 is a natural number) is also found in terms of the polynomials parameters by means of two error-free computing approaches; one makes use of the Lauricella functions, which is based on the Srivastava-Niukkanen 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 logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by 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 quasi-linear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for n>>1).