Z. Huang, T. Huber, J. McLaughlin, P. Wang, D. Ye and Y. Xu, Sign-patterns of Certain Infinite Products, Annals of Combinatorics (2026).

Project description

In this project, we studied and characterized the sign distribution of the coefficients of the expansion of certain infinite products. For example, for a prime $p>3$ and a positive integer $i$ not divisible by $p$, if we write \(\prod_{n=1}^{\infty}\frac{(1-q^{in})}{(1-q^{pn})}=\sum_{n=0}^{\infty}a_{n}^{[i,p]}q^{n},\) then we proved that there is a constant $M$ that can be explicitly formulated in terms of $p$ and $i$ such that for $n>M$, % When $gcd(i,m)=1$:
\begin{enumerate} \item if $p \equiv 1 \pmod 3 $, \begin{align} a_n^{[i,p]}\begin{cases} >0, &\mbox{if $n\equiv i(6r^2+r) \pmod p$ with $r\leq \frac{4p-1}{12}$ or $r>\frac{10p-1}{12}$,}
<0, &\mbox{if $n\equiv i(6r^2+r) \pmod p$ with $\frac{4p-1}{12} <r\leq \frac{10p-1}{12}$,}
=0, &\mbox{if $n\neq i(6r^2+r) \pmod p$,} \end{cases} \end{align} \item if $p \equiv -1 \pmod 3 $, then \begin{align} a_n^{[i,p]}\begin{cases} >0, &\mbox{if $n\equiv i(6r^2+r) \pmod p$ with $r\leq \frac{2p-1}{12}$ or $r>\frac{8p-1}{12}$,}
<0, &\mbox{if $n\equiv i(6r^2+r) \pmod p$ with $\frac{2p-1}{12} <r\leq \frac{8p-1}{12}$,}
=0, &\mbox{if $n\neq i(6r^2+r) \pmod p$.} \end{cases} \end{align} \end{enumerate}