Nonmodular infinite products and a conjecture of Seo and Yee

We will tackle a conjecture of S. Seo and A. J. Yee, which says that the series expansion of $1/(q,-q^3;q^4)_\infty$ has nonnegative coefficients. Our approach relies on an approximation of the generally nonmodular infinite product $1/(q^a;q^M)_\infty$, where $M$ is a positive integer and $a$ is any of $1,2,\ldots,M$.