From Philippe.Flajolet@inria.fr Wed Jul 23 20:27:07 1997
Date: Wed, 23 Jul 1997 20:27:05 +0200
From: Philippe Flajolet <Philippe.Flajolet@inria.fr>
Subject: Zeros of polynomials



This problem is interesting. More generally, if a polynomial $P_n(x)$
with nonnegative coefficients has all its zeros that are real negative
then, a host of properties follow and they apply in particular to the
probability distribution with PGF $P_n(x)/P_n(1)$. The mean $\mu_n$
and the standard deviation $\sigma_n$ play a special r\^ole.

        ---The distribution is unimodal (and so are the coefficients
of $P(x)$).
        -- The maximum is at a small distance (<=1) from the mean and thus
well ``localized''. [Benhoumani, PhD Lyon, 1993; must have appeared
somewhere!] An old conjecture of Schinzel, if I remember correctly.
        -- If $\sigma_n$ tends to infinity, then the coefficients
obeys a LOCAL limit Gaussian distribution of the Gaussain type.
[Harper's lemma, see, eg, Bender, Central and Local Limit Theorems,
JCTA, 1973]. There's probably something about speed of distribution
that might result from Berry-Esseen.

Typical cases are Stirling 1st kind (obvious) and Stirling 2nd kind
(Harper's original proof of normality).

===>    It'd be nice if somebody had the energy to write a 2--3 page
summary for these pages. [I am not an expert]

        Cheers, Philippe