Further results on vanishing coefficients in the series expansion of lacunary eta quotients

Abstract: For a function \(A(q)=\sum_{n\geq 0} a_n q^n\), define $A_{(0)}:={n\in \mathbb{N}: a_n=0}.$ If $A(q)$ and $B(q)$ satisfy $A_{(0)}=B_{(0)}$, then we say that $A(q)$ and $B(q)$ have \emph{identically vanishing coefficients}.

In a previous paper the authors proved the existence of various pairs $(A(q),B(q))$ of lacunary eta quotients with identically vanishing coefficients. The work in that previous paper was motivated by a result of Han and Ono, who showed that $f_1^8$ and $f_3^3/f_1$ have identically vanishing coefficients (here \(f_i = \prod_{n=0}^{\infty}(1-q^{in})\)). In each of these pairs, one of the eta quotients was a power of $f_1$, whose lacunarity was described in a paper by Serre.

Further experiments indicated that the results in this previous paper were just the ``tip of the iceberg’’. In the current work, we demonstrate that there is a much larger list of eta quotients $B(q)$ having coefficients that vanish identically with those of $A(q)=f_1^6$. Similar results hold for $f_1^r$, $r=4, 8, 10, 14, 26$, and for $f_1^3f_2^3$. A natural network structure exists on the set of eta quotients $C(q)$ for which $A_{(0)}\subsetneqq C_{(0)}$. The network may be exhibited by partially ordering the sets of vanishing coefficients by inclusion and constructing a directed graph of collections of eta quotients that have identically vanishing coefficients.

We provide a comprehensive description of what experiment suggests and employ a variety of methods to prove experimentally-derived results. The work is a template and atlas for the subsequent study of lacunary eta quotients. A broad range of proof strategies are applied to confirm the vanishing structure. These comprise a representative sample of techniques that may be used to study the remaining observations and conjectures resulting from the work.