Vanishing Coefficients in Products of Quintuple Products

Explicit arithmetic progressions modulo primes $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ are derived in which the coefficients in the expansions of products of quintuple products vanish. In particular, if $p=m^2+n^2$, and $b$ is a positive integer, and $$\sum _{n=0}^{\mathrm{\infty }}{a}_n{q}^n=\frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p{)}_{\mathrm{\infty }}}{(q^p,-q^{bm},-q^{p-bm},-q^{bn},-q^{p-bn};q^p{)}_{\mathrm{\infty }}^2},$$ we determine $\alpha =\alpha (m,n,p)$ such that $a_{pt+\alpha }=0$. Our results are proven using involutive transformations on integer lattices.