Positivity of Szegö's rational function

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

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