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, (Wayback Machine pdf)


The answer is yes.

This was already proved in Gabor Szegö's original paper 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.


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