Direct spreading measures of Laguerre polynomials

The direct spreading measures of the Laguerre polynomials L (α) n (x), which quantify the distribution of its Rakhmanov probability density ρn,α(x) = 1 dn xαe−x [ L (α) n (x) ]2 along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar rootmean-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 F (2q+1) A ( 1 q , ..., 1 q ; 1 ) , 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).