Proof of the Andrews-El Bachraoui positivity conjecture

TL;DR

This paper proves a conjecture by Andrews and El Bachraoui that all coefficients in a specific q-series expansion are positive, using q-hypergeometric methods, thereby confirming their recent mathematical conjecture.

Contribution

It provides a rigorous proof of the Andrews-El Bachraoui positivity conjecture for a particular q-series using q-hypergeometric techniques.

Findings

Confirmed all coefficients in the series are positive.

Validated the conjecture for all k ≥ 1.

Demonstrated the effectiveness of q-hypergeometric methods.

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.