Séminaire Lotharingien de Combinatoire, B05e (1981).
[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+4xyz)-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 paper has been finally published under the same title in SIAM J. Math. Anal. 14 (1983), 396-398.