Proof of the Andrews-El Bachraoui positivity conjecture
Shane Chern, Chun Wang
TL;DR
This work resolves the Andrews–El Bachraoui positivity conjecture by showing that for every positive integer $k$, the $q$-series $F_{k,1}(q)=\sum_{n\ge0} \frac{(q^{2n+2}, q^{2n+2k}; q^2)_\infty}{(q^{2n+1};q^2)_\infty^2} q^{2n}$ has all positive coefficients. The authors develop a self-contained method, dubbed the 'substitution of one' trick, to decompose $\frac{(q^2;q^2)_n}{(q;q^2)_n^2}$ into a nonnegative sum of $q$-binomial terms, and then leverage $q$-hypergeometric transformations (notably ${}_2\phi_1$, Heine’s transformation, and Fine’s transformation) to reformulate $F_{k,1}(q)$ into explicit nonnegative sums. A key corollary establishes the positivity of $(q^2;q^2)_n/(q;q^2)_{n+1}^2$, which underpins all subsequent decompositions. Consequently, for $k=1$ one recovers a simple positive series, and for $k\ge2$ a finite nonnegative decomposition in terms of ${k-2\brack n}_{q^2}$ yields positivity for all $k$, thereby proving the conjecture. The result streamlines previous case-by-case analyses and reinforces the role of hypergeometric transformations in establishing positivity of partition-generating functions.
Abstract
We prove that for $k\ge 1$, all coefficients in the expansion of the series $$\sum_{n\ge 0} \frac{(q^{2n+2}, q^{2n+2k}; q^2)_\infty}{(q^{2n+1};q^2)_\infty^2} q^{2n}$$ are positive, by $q$-hypergeometric means. This confirms a recent conjecture of Andrews and El Bachraoui.