Vanishing Coefficients in Products of Quintuple Products

Abstract

Explicit arithmetic progressions modulo primes $p \equiv 1 \pmod{4}$ 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 $$\sum_{n=0}^{\infty} a_{n}q^{n} = \frac{(q^{2bm},q^{p-2bm};q^{2bn},q^{p-2bn};q^p)_{\infty}}{(q^p,-q^{b m},-q^{p-bm},-q^{bn},-q^{p-bn};q^p)_{\infty}^2},$$ we determine $α= α(m,n,p)$ such that $a_{pt+ α}=0$. Our results are proven using involutive transformations on integer lattices.