关于无穷乘积展开式中消失系数的一些结果
Some results on vanishing coefficients in infinite product expansions
摘要：
最近，M。D. Hirschhorn证明，如果$\sum_{n=0}^\infty a_nq^n := (-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3$和$\sum_{n=0}^\infty b_nq^n:=(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3$，那么$a_{5n+2}=a_{5n+4}=0$和$b_{5n+1}=b_{5n+4}=0$。受Hirschhorn，D. Tang证明了一些可比较的结果，包括：如果$ \sum_{n=0}^\infty c_nq^n := (-q,-q^4;q^5)_\infty^3(q^3,q^7;q^{10})_\infty$和$\sum_{n=0}^\infty d_nq^n := (-q^2,-q^3;q^5)_\infty^3(q,q^9;q^{10})_\infty$，则$c_{5n+3}=c_{5n+4}=0$和$d_{5n+3}=d_{5n+4}=0$。本文证明了$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}$，和$c_{5n+1}>d_{5n+1}$ .我们还记录了Tang未列出的其他一些可比结果。
英文摘要：
Recently, M. D. Hirschhorn proved that, if $\sum_{n=0}^\infty a_nq^n := (-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3$ and $\sum_{n=0}^\infty b_nq^n:=(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3$, 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_nq^n := (-q,-q^4;q^5)_\infty^3(q^3,q^7;q^{10})_\infty$ and $\sum_{n=0}^\infty d_nq^n := (-q^2,-q^3;q^5)_\infty^3(q,q^9;q^{10})_\infty$, 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.
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