Some results on vanishing coefficients in infinite product expansions

Abstract

Recently, M. D. Hirschhorn proved that, if ${\sum }_{n=0}^{\infty }{a}_n{q}^n:=(-q,-q^4;q^5{)}_{\infty }(q,q^9;q^{10}{)}_{\infty }^3$ and ${\sum }_{n=0}^{\infty }{b}_n{q}^n:=(-q^2,-q^3;q^5{)}_{\infty }(q^3,q^7;q^{10}{)}_{\infty }^3$, then $a_{5n+2}=a_{5n+4}=0$ and $b_{5n+1}=b_{5n+4}=0$. Motivated by the work of Hirschhorn, D. Tang proved some comparable results including the following: If ${\sum }_{n=0}^{\infty }{c}_n{q}^n:=(-q,-q^4;q^5{)}_{\infty }^3(q^3,q^7;q^{10}{)}_{\infty }$ and ${\sum }_{n=0}^{\infty }{d}_n{q}^n:=(-q^2,-q^3;q^5{)}_{\infty }^3(q,q^9;q^{10}{)}_{\infty }$, then $c_{5n+3}=c_{5n+4}=0$ and $d_{5n+3}=d_{5n+4}=0$. In this paper, we prove that $a_{5n}=b_{5n+2}$, $a_{5n+1}=b_{5n+3}$, $a_{5n+2}=b_{5n+4}$, $a_{5n-1}=b_{5n+1}$, $c_{5n+3}=d_{5n+3}$, $c_{5n+4}=d_{5n+4}$, $c_{5n}=d_{5n}$, $c_{5n+2}=d_{5n+2}$, and $c_{5n+1}>d_{5n+1}$. We also record some other comparable results not listed by Tang.

15 pages