Generalizations of Hirschhorn’s Results on Two Remarkable q-Series Expansions

Abstract

Recently, Hirschhorn investigated vanishing coefficients of the arithmetic progressions in the following two q-series expansions $$\begin{array}{c}\sum _{n=0}^{\infty }a(n)q^n:=\prod _{n=1}^{\infty }(1+q^{5n-4})(1+q^{5n-1}){(1-q^{10n-9})}^3{(1-q^{10n-1})}^3,\\ \sum _{n=0}^{\infty }b(n)q^n:=\prod _{n=1}^{\infty }(1+q^{5n-3})(1+q^{5n-2}){(1-q^{10n-7})}^3{(1-q^{10n-3})}^3.\end{array}$$

He proved that for $n\ge 0,a(5n+2)=a(5n+4)=b(5n+1)=b(5n+4)=0.$ In this paper, we further study these two q-series expansions and obtain the generating functions of $a(10n+r)$ and $b(10n+r)$ $(0\le r\le 9)$ by using two MAPLE packages, qseries and thetaids, due to Jie Frye and Frank Garvan. The signs of $a(10n+r)$ and $b(10n+r)$ are determined, which imply Hirschhorn’s results given above.

Keywords:

• q-series expansions
• vanishing coefficients
• signs

AMS Subject Classification:

• 33D15
• 11F33
• 30B10

Additional information

Funding

The first author was supported by the National Natural Science Foundation of China (Nos. 11571143, 11971203), the Natural Science Foundation of Jiangsu Province of China (No. BK20180044). The second author was supported by the National Natural Science Foundation of China (No. 11901430).