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.

摘要

最近，Hirschhorn 在以下两个q系列展开中研究了算术级数的消失系数$$\begin{array}{c}\sum _{n=0}^{\infty }\mathrm{一个}(n)q^n：=\prod _{n=1}^{\infty }(1+q^{5n-4})(1+q^{5n-1}){(1-q^{10n-9})}^3{(1-q^{10n-1})}^3,\\ \sum _{n=0}^{\infty }b(n)q^n：=\prod _{n=1}^{\infty }(1+q^{5n-3})(1+q^{5n-2}){(1-q^{10n-7})}^3{(1-q^{10n-3})}^3.\end{array}$$

他证明了对于$n\ge 0,\mathrm{一个}(5n+2)=\mathrm{一个}(5n+4)=b(5n+1)=b(5n+4)=0.$在本文中，我们进一步研究了这两个q级数展开并获得了$\mathrm{一个}(10n+r)$和$b(10n+r)$ $(0\le r\le 9)$由于 Jie Frye 和 Frank Garvan ，使用了两个 MAPLE 包qseries和thetaids 。的迹象$\mathrm{一个}(10n+r)$和$b(10n+r)$是确定的，这意味着上面给出的 Hirschhorn 的结果。