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\rm
\bf
A Generalization of Odlyzko's Conjecture: The Coefficients of 
$  \, \, { (1 - q)^j / ((1- q^{2n} ) ... (1- q^{2n+2j} ))} \, \,$ 
Alternate in Sign\rm

{\it Jane Friedman and Doron Zeilberger}

{\bf Appeared in:}   Proc. Amer. Math. Soc. {\bf 118}, 1013 (1993).

Let $N,a,b,i,j,r,n$ denote arbitrary non-negative integers, and let

$$
  (x)_N :=  \prod_{i=0}^{N-1} (1- q^i x )  \, \, \,, G(a,b):= 
{{(q)_{a+b}} \over{ (q)_a (q)_b }} \,,  \, \,
A:= \{f(q) \,; \, (-1)^i f^{(i)} (0) \geq 0, 0 \leq i < \infty \}. 
$$

A. Odlyzko conjectured that $(1-q)^j /(q)_j  \epsilon  A$.
This was proved in [1], 
where the more general
fact that $(1-q)^j G(2j,r)  \epsilon  A$ was proved. Here
we show that this last result also implies 
the statement of the title, that generalizes Odlyzko's
conjecture (for even $j$) in a different direction. To wit: 
set $t= q^{2n} , N=2j$, in the q-binomial theorem
$1/ (t)_{N+1} = \sum_{r>= 0} G(N,r) t^r $, and then multiply
by $(1-q)^j$. QED.

{\bf Reference}

1. D.Stanton and D. Zeilberger, \it
The Odlyzko conjecture and O'Hara's
unimodality proof\rm
, Proc. Amer. Math. Soc. {\bf 107}(1989),
39-42.

Department of Mathematics, University of San Diego, Alcala Park, San
Diego, CA 92210.

Department of Mathematics, Temple University, Philadelphia, PA19122.

\_\_\_\_\_\_\_\_\_

{\it 1980 Mathematics Subject Classification (1985 Revision)}. Primary
05A17,10A45.

The second author was supported by NSF grant DMS8901690.
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