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Séminaire Lotharingien de Combinatoire, B05e (1981), 11 pp.

[Formerly: Publ. I.R.M.A. Strasbourg, 1982, 182/S-04, p.
84-94.]

# Joe Gillis, Bruce Reznick and Doron Zeilberger

# On Elementary Methods in Positivity Theory

**Abstract.**
We raise conjectures concerning the positivity of the power series
coefficients of multi-variate rational functions. We also give
a short proof of a result of Askey and Gasper that
(1-*x-y-z*+4*xyz*)^{-b} has positive power series
coefficients for *b*>=(*\sqrt{17}*-3)/2. We show how
Ismail and Tanharkar's proof that

(1-(1-*L*)*x*-*Ly-Lxz*-(1-*L*)*yz+xyz*)^{-a}
(where 0<=*L*<=1)
has positive power series coefficients for *a*=1 implies
Koornwinder's result that it does so for *a*>=1.

The following version is available:

The paper has been finally published under the same title in
*SIAM J. Math. Anal.* **14** (1983), 396-398.