Andrews–Merca–Guo–Zeng truncated Jacobi triple product positivity conjecture

Let $R$, $S$, and $k$ be integers with $k\geq 1$ and $1\leq S\leq R/2$. For $n\geq 1$, consider the coefficient of $q^n$ in

$$\frac{(-1)^{k-1}}{(q^S,q^{R-S},q^R;q^R)_\infty}\sum_{j=-k+1}^{k}(-1)^jq^{Rj(j-1)/2+Sj}.$$

Andrews–Merca–Guo–Zeng conjecture. This coefficient is nonnegative.

The conjecture proposes a positivity property for truncations of the Jacobi triple product, extending truncated identities studied by Andrews and Merca and by Guo and Zeng. It was proved independently by Mao and Yee, and later reconfirmed by Wang and Yee through an explicit series form with nonnegative coefficients.