Dissections of lacunary eta quotients and identically vanishing coefficients

Abstract: For any function $A(q)= \sum_{n=0}^{\infty}a_nq^n$ define [ A_{(0)}:={n\in \mathbb{N}: a_n=0}. ] Now suppose $C(q)$ and $D(q)$ are two functions whose $m$-dissections are given by \begin{align} C(q)& = c_0G_0(q^m)+c_1 q G_1(q^m)+ \dots + c_{m-1} q^{m-1} G_{m-1}(q^m),
D(q)& = d_0G_0(q^m)+d_1 q G_1(q^m)+ \dots + d_{m-1} q^{m-1} G_{m-1}(q^m). \notag \end{align} If it is the case that $c_i=0\lrla d_i=0$, $i=0,1,\dots, m-1$, then we say that $C(q)$ and $D(q)$ have \emph{similar $m$-dissections}, and then it is also clear that $C_{(0)} = D_{(0)}$, in which case we say that $C(q)$ and $D(q)$ have \emph{identically vanishing coefficients}.

In the present paper some new 4-dissections of particular eta quotients are developed. These are used in conjunction with known 2- and 3-dissections to prove many results on the identical vanishing of coefficients of various eta quotients, results which were found experimentally and partially proved in another paper by the present authors.

Similar arguments allow many results of the form $C_{(0)} \subsetneqq D_{(0)}$ to be proved for many pairs of lacunary eta quotients $C(q)$ and $D(q)$.