Dissection of the quintuple product, with applications
Published in Journal of Combinatorial Theory, Series A, 2026
Abstract: This work considers the $m$-dissection (for $m\not \equiv 0 \pmod 3$) of the general quintuple product
\[Q(z,q)= (z,q/z,q;q)_{\infty}(q z^2,q/z^2;q^2)_{\infty}.\]Multiple novel applications arise from this $m$-dissection. For example, we derive the general partition identity
\[D_S(m n+(m^2-1)/24) = (-1)^{(m+1)/6} b_m(n),\hspace{25pt} \text{ for all }n \geq 0,\]where $m\equiv5\pmod{6}$ is a square-free positive integer relatively prime to~$6$; $D_S(n)$ is defined, for $S$ the set of positive integers containing no multiples of~$m$, to be the number of partitions of $n$ into an \underline{even} number of distinct parts from $S$ minus the number of partitions of $n$ into an \underline{odd} number of distinct parts from $S$; and $b_m(n)$ denotes the number of $m$-regular partitions of $n$. The dissections allow us to prove a conjecture of Hirschhorn concerning the $2^n$-dissection of $(q;q){\infty}$, as well as determine the pattern of the sign changes of the coefficients $a{n}$ of the infinite product
\[\frac{(q^{2^{k-1}};q^{2^{k-1}})_{\infty }}{(q^{p};q^{p})_{\infty}^{2}}=\sum_{n=0}^{\infty}a_{n}q^{n} , \quad k \ge 1,\quad p \ge 5\ \text{a prime.}\]This covers a recent result of Bringmann et al. that corresponds to the case $k=1$ and $p=5$.