From e8925428@student.tuwien.ac.at Wed Jul 23 15:20:25 1997
Date: Wed, 23 Jul 1997 15:20:20 +0200 (METDST)
From: Bernhard Gittenberger <e8925428@student.tuwien.ac.at>
Subject: An open problem (from Hsien-Kuei Hwang)


Define the sequence of probability generating functions by
$g_0(z)=$ and for $n\ge1$
\[
    g_n(z) = z\sum_{0\le j\le n-1} \frac{(n-j)n!}
    {n^{n-j+1}j!}\,g_j(z).
\]
Show that the zeros of $g_n(z)$ (as a polynomial of degree
$n$) are all real (and thus non-positive). This sequence
of pgf arises from algorithmic analysis of some marketing
problems; see Frieze and Pittel (1995), Ann. App. Prob. 
Also the terms in the summation besides $g_j(z)$ define
a distribution called Naor distribution (see Johnson et
al. Univariate Discrete Distributions). This distribution
is also related to the number of cyclic points in random 
mappings (see Kolchin, Random Mappings). 

Amities
Hsien-Kuei