Positivity of Szegö's rational function

This article has been published in Advances in Applied Mathematics (Volume 41, Issue 2, August 2008, Pages 255-264) and is available at doi:10.1016/j.aam.2007.10.001.

Errata and addenda are contained in an additional document.

Abstract

We consider the problem of deciding whether a given rational function has a power series expansion with all its coefficients positive. Introducing an elementary transformation that preserves such positivity we are able to provide an elementary proof for the positivity of Szegö's function

$\frac{1}{(1-x)(1-y)+(1-y)(1-z)+(1-z)(1-x)}$

which has been at the historical root of this subject starting with Szegö. We then demonstrate how to apply the transformation to prove a 4-dimensional generalization of the above function, and close with discussing the set of parameters (a,b) such that

$\frac{1}{1-(x+y+z)+a(xy+yz+zx)+bxyz}$