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\centerline
{\bf Investigating the Sahi Property }
\bigskip
\centerline
{\it Yukun Yao and Doron ZEILBERGER}

In [S], Lemma 16, it is proved, using ad-hoc, human methods, that if
$\alpha, \beta, \gamma, \delta$, are non-negative and $\alpha+\beta+\gamma+\delta=1$  and $\alpha\,\delta \geq \beta \, \gamma$, then
the following formal power series
$$
1-(1-a)^\delta \, (1-a-b-c)^\gamma \, (1-a-b-d)^\beta \, (1-a-b-c-d-e)^\alpha \quad ,
$$

is a formal power series in the variables $a,b,c,d,e$ with {\bf non-negative} coefficients.

For now, for the sake of simplicity, let's take $\alpha=\beta=\gamma=\delta=\frac{1}{4}$, so we have, thanks to Sahi the fact that
$$
1-((1-a) (1-a-b-c) (1-a-b-d) (1-a-b-c-d-e))^{\frac{1}{4}} \quad ,
$$
is a formal power series in the variables $a,b,c,d,e$ with {\bf non-negative} coefficients.

Let $x_1, \dots, x_n$ be formal variables and let $S=\{ \sum_{j=1}^{n} c_{ij}x_i,i=1..m)\}$ be a set of
linear combinations  of the variables. Let $\alpha$ be a number, the pair $(S,\alpha)$ has the $(n,m)$-{\bf Sahi property} if
the formal power series
$$
1- \prod_{1}^{m}(1- \sum_{j=1}^{n} c_{ij}x_i)^\alpha \quad,
$$
is a formal power series with {\bf non-negative} coefficients.

So Sahi's Lemma 16 says that the  pair $(\{x_1,x_1+x_2+x_3,x_1+x_2+x_4,x_1+x_2+x_3+x_4+x_5\},\frac{1}{4})$ has
the $(5,4)$-Sahi property.


{\bf Big Challenge} For given $m$ and $n$ find, experimentally, and if possible rigorously, all
pairs $(S,\alpha)$ with the Sahi property.

We have to start with the first not completely trivial case, $n=1$ and $m=2$.

using the Maple program {\tt SahiProperty.txt} it is seen that
$\{x,x/4\},\frac{7}{10})$ does {\bf not} have the Sahi property, but $\{x,x/4\},\frac{3}{5})$ does.


{\bf First mini-challenge}

Find all pairs $(c\alpha)$ such that the one-variable formal power series
$$
1-((1-x)\,(1-c\,x))^{\alpha} \quad,
$$
is a formal power series with non-negative coefficients.


\bigskip
{\bf References}
[S] Siddhartha Sahi, {\it Higher correlation inequalities}, Combinatorica {\bf 28} (2008), 209-227.
\bigskip
\hrule
\bigskip
Yukun Yao, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt yao at math dot rutgers dot edu}   \quad .
\bigskip
Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt DoronZeil at gmail  dot com}   \quad .


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