Dissections of lacunary eta quotients and identically vanishing coefficients

Abstract

For any function $A\left(q\right)={\sum }_{n=0}^{\infty }{a}_n{q}^n$ define$A_{\left(0\right)}:=\{n\in N:a_n=0\}.$ Now suppose $C\left(q\right)$ and $D\left(q\right)$ are two functions whose m-dissections are given by$C\left(q\right)=c_0{G}_0\left(q^m\right)+c_{1}q{G}_1\left(q^m\right)+\dots +c_{m-1}{q}^{m-1}{G}_{m-1}\left(q^m\right),$$D\left(q\right)=d_0{G}_0\left(q^m\right)+d_{1}q{G}_1\left(q^m\right)+\dots +d_{m-1}{q}^{m-1}{G}_{m-1}\left(q^m\right).$ If it is the case that $c_i=0⟺d_i=0$, $i=0,1,\dots ,m-1$, then we say that $C\left(q\right)$ and $D\left(q\right)$ have similar m-dissections, and then it is also clear that $C_{\left(0\right)}=D_{\left(0\right)}$, in which case we say that $C\left(q\right)$ and $D\left(q\right)$ have 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 $C_{\left(0\right)}⫋D_{\left(0\right)}$ to be proved for many pairs of lacunary eta quotients $C\left(q\right)$ and $D\left(q\right)$.