Dissection of the quintuple product, with applications

Abstract

This work considers the m-dissection (for $m\not\equiv 0\left(\mathrm{mod}3\right)$) 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$D_S(mn+(m^2-1)/24)={(-1)}^{(m+1)/6}{b}_m\left(n\right),\text{ for all }n\ge 0,$ where $m\equiv 5\left(\mathrm{mod}6\right)$ is a square-free positive integer relatively prime to 6; $D_S\left(n\right)$ is defined, for S the set of positive integers containing no multiples of m, to be the number of partitions of n into an even number of distinct parts from S minus the number of partitions of n into an odd number of distinct parts from S; and $b_m\left(n\right)$ 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$\frac{{(q^{2^{k-1}};q^{2^{k-1}})}_{\infty }}{{(q^p;q^p)}_{\infty }^2}=\sum _{n=0}^{\infty }{a}_n{q}^n,k\ge 1,p\ge 5\text{a prime.}$ This covers a recent result of Bringmann et al. that corresponds to the case $k=1$ and $p=5$.