Investigating the Sahi Property Yukun Yao and Doron ZEILBERGER

In [S], Lemma 16, it is proved, using ad-hoc, human methods, that if α, β, γ, δ, are non-negative and α + β + γ + δ = 1 and α δ ≥ β γ, then the following formal power series

1 − (1 − a)^{δ} (1 − a − b − c)^{γ} (1 − a − b − d)^{β} (1 − a − b − c − d − e)^{α} ,

is a formal power series in the variables a, b, c, d, e with non-negative coefficients. For now, for the sake of simplicity, let’s take α = β = γ = δ = \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}} ,

is a formal power series in the variables a, b, c, d, e with non-negative coefficients. Let x_{1}, …, x_{n} be formal variables and let S = {∑_{j=1}^{n} c_{ij}x_{i}, i = 1..m)} be a set of linear combinations of the variables. Let α be a number, the pair (S, α) has the (n, m)-Sahi property if the formal power series

1 − ∏_{1}^{m}(1 − ∑_{j=1}^{n} c_{ij}x_{i})^{α} ,

is a formal power series with 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. Big Challenge For given m and n find, experimentally, and if possible rigorously, all pairs (S, α) with the Sahi property. We have to start with the first not completely trivial case, n = 1 and m = 2. using the Maple program SahiProperty.txt it is seen that \{x, x/4\}, \frac{7}{10}) does not have the Sahi property, but \{x, x/4\}, \frac{3}{5}) does. First mini-challenge Find all pairs (cα) such that the one-variable formal power series

1 − ((1 − x) (1 − c x))^{α} ,

is a formal power series with non-negative coefficients.

References [S] Siddhartha Sahi, Higher correlation inequalities, Combinatorica 28 (2008), 209-227.

Yukun Yao, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: yao at math dot rutgers dot edu .

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: DoronZeil at gmail dot com .