Some results on vanishing coefficients in infinite product expansions

Abstract

Recently, Hirschhorn proved that, if

$$\begin{aligned} \sum _{n=0}^{\infty }a_nq^n&:=(-\,q,-\,q^4;q^5)_\infty (q,q^9;q^{10})_\infty ^3 \end{aligned}$$

and

$$\begin{aligned} \sum _{n=0}^{\infty }b_nq^n:=(-\,q^2,-\,q^3;q^5)_\infty (q^3,q^7;q^{10})_\infty ^3, \end{aligned}$$

then $a_{5n+2}=a_{5n+4}=0$ and $b_{5n+1}=b_{5n+4}=0$. Motivated by the work of Hirschhorn, Tang proved some comparable results including the following:

If

$$\begin{aligned} \sum _{n=0}^{\infty }c_nq^n&:=(-\,q,-\,q^4;q^5)_\infty ^3(q^3,q^7;q^{10})_\infty \end{aligned}$$

and

$$\begin{aligned} \sum _{n=0}^{\infty }d_nq^n:=(-\,q^2,-\,q^3;q^5)_\infty ^3(q,q^9;q^{10})_\infty , \end{aligned}$$

then

$$\begin{aligned} c_{5n+3}=c_{5n+4}=0\,\,\, \mathrm{{and}}\,\,\, d_{5n+3}=d_{5n+4}=0. \end{aligned}$$

In this paper, we prove that

$$\begin{aligned} a_{5n}&=b_{5n+2},~a_{5n+1}=b_{5n+3},~a_{5n+2}=b_{5n+4},~ a_{5n-1}=b_{5n+1},\\ c_{5n+3}&=d_{5n+3},~ c_{5n+4}=d_{5n+4},~ c_{5n}=d_{5n},~ c_{5n+2}=d_{5n+2}~ and ~ c_{5n+1}>d_{5n+1}, \end{aligned}$$

We also record some other comparable results not listed by Tang.