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))