A rational function is called positive if all its Taylor coefficients are positive.
 Friedrichs-Lewy conjecture states the positivity of the rational function
 \begin{eqnarray\*}\frac{1}{
(1-x)(1- y)+(1- y)(1-z)+(1-z)(1-x)}
= \sum\limits\_{ k,m,n\ge0}
a\_{k,m, n }x^k y^mz^n. \end{eqnarray\*}
The conjecture was first proved by G. Szego. 


Let $P\_n=\prod\limits\_{i=1}^n(1-x\_i)$, is it true that the following rational function is positive \begin{eqnarray\*}\frac{1}{
\sum\limits\_{i=1}^n\frac{P\_n}{1-x\_i}}=\sum\limits\_{i\_1,i\_2\cdots, i\_n\ge 0}a\_{i\_1,i\_2\cdots, i\_n }x\_1^{i\_1}x\_2^{i\_2}\cdots x\_n^{i\_n}\quad? \end{eqnarray\*}


A reference:



> 
> Armin Straub, *Positivity of Szegö's rational function*, Advances in Applied Mathematics
>  **41** Issue 2 (2008) pp 255–264, doi:[10.1016/j.aam.2007.10.001](https://doi.org/10.1016/j.aam.2007.10.001), ([Wayback Machine pdf](http://web.archive.org/web/20170922235432/http://129.81.170.14/~tamdeberhan/positivity.pdf))
> 
> 
> 



Positivity of a rational function
The answer is yes.


This was already proved in Gabor Szegö's [original paper](http://www.springerlink.com/content/tgmq242t51414177/) from 1933:


G. Szegö, *Über gewisse Potenzreihen mit lauter positiven Koeffizienten*, Mathematische Zeitschrift, Volume 37, Number 1, 674-688, DOI: 10.1007/BF01474608


The result can be found in Paragraph 3 "Verallgemeinerungen". However, apparently the simplified proof method due to Armin Straub which you mention applies also to this generalization.


