Lacunary eta quotients with identically vanishing coefficients

Abstract: For \(|q|<1\), define \(f_i=\prod_{n=1}^{\infty} (1-q^{in})\), and let \((A(q), B(q))\) be any of the pairs \(\left(f_1^{4},\frac{f_1^8}{f_2^2}\right),\) \(\left(f_1^{4},\frac{f_1^{10}}{f_3^2}\right),\) \(\left(f_1^{6},\frac{f_2^4}{f_1^2}\right),\) \(\left(f_1^{6},\frac{f_1^{14}}{f_2^4}\right),\) \(\left(f_1^{10},\frac{f_2^6}{f_1^2}\right),\) \(\left(f_1^{14},\frac{f_3^5}{f_1}\right), \left(f_1^{14},\frac{f_2^8}{f_1^2}\right).\) For any such pair \((A(q), B(q))\), define the sequences \(\{a(n)\}\) and \(\{b(n)\}\) to be the coefficients of \(q^{n}\) of \(A(q)\) and \(B(q)\), respectively. Then for each pair it is shown that \(a(n)\) vanishes if and only if \(b(n)\) vanishes. In each case, a criterion is given which states precisely when \(a(n)=b(n)=0\). Moreover, for the pairs \(\left(f_1^{26},\frac{f_3^9}{f_1}\right),\) \(\left(f_1^{26},\frac{f_2^{16}}{f_1^6}\right)\) it is shown that \(a(n)=b(n)=0\) if $12n+13$ satisfies a criteria of Serre for \(a(n)=0\).