GustafsonによるC_n型Baileyの6ψ6和公式
前の記事( Rosengrenによる$C_n$型Jacksonの和公式 )の記法を用いる.
$C_n$型Jacksonの和公式
Rosengrenによる$C_n$型Jacksonの和公式 において$p=0$とすると以下を得る.
非負整数$m_1,\dots,m_n$に対し, $a^2q^{1+m_1+\cdots+m_n}=bcde$であるとき,
\begin{align}
&\sum_{y_1,\dots,y_n=0}^{m_1,\dots,m_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-az_jz_kq^{y_j+y_k}}{1-az_jz_k}\\
&\qquad\cdot\prod_{1\leq j,k\leq n}\frac{(az_jz_k,z_kq^{-m_j}/z_j;q)_{y_k}}{(z_kq/z_j,az_jz_kq^{m_j+1};q)_{y_k}}\\
&\qquad\cdot\prod_{k=1}^n\frac{(bz_k,cz_k,dz_k,ez_k;q)_{y_k}}{(az_kq/b,az_kq/c,az_kq/d,az_kq/e;q)_{y_k}}q^{y_k}\\
&=\frac{\prod_{1\leq j,k\leq n}(az_jz_kq;q)_{m_k}}{\prod_{1\leq j< k\leq n}(az_jz_kq;q)_{m_j+m_k}}\frac{(aq/bc,aq/bd,aq/cd;q)_{m_1+\cdots+m_n}}{\prod_{k=1}^n(az_kq/b,az_kq/c,az_kq/d,eq^{-m_k}/az_k;q)_{m_k}}
\end{align}
が成り立つ.
$C_n$型Rogersの和公式
定理1において, $e=a^2q^{1+m_1+\cdots+m_n}/bcd$を代入して, $d$以外を固定して$d\to\infty$とすると, 以下の系を得る.
\begin{align}
&\sum_{y_1,\dots,y_n=0}^{m_1,\dots,m_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-az_jz_kq^{y_j+y_k}}{1-az_jz_k}\\
&\qquad\cdot\prod_{1\leq j,k\leq n}\frac{(az_jz_k,z_kq^{-m_j}/z_j;q)_{y_k}}{(z_kq/z_j,az_jz_kq^{m_j+1};q)_{y_k}}\\
&\qquad\cdot\prod_{k=1}^n\frac{(bz_k,cz_k;q)_{y_k}}{(az_kq/b,az_kq/c;q)_{y_k}}\left(\frac{aq^{m_1+\cdots+m_n+1}}{bc}\right)^{y_k}\\
&=\frac{\prod_{1\leq j,k\leq n}(az_jz_kq;q)_{m_k}}{\prod_{1\leq j< k\leq n}(az_jz_kq;q)_{m_j+m_k}}\frac{(aq/bc;q)_{m_1+\cdots+m_n}}{\prod_{k=1}^n(az_kq/b,az_kq/c;q)_{m_k}}
\end{align}
が成り立つ.
これは
\begin{align}
&\sum_{0\leq y_1,\dots,y_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-az_jz_kq^{y_j+y_k}}{1-az_jz_k}\\
&\qquad\cdot\prod_{1\leq j,k\leq n}\frac{(az_jz_k,z_k/t_jz_j;q)_{y_k}}{(z_kq/z_j,at_jz_jz_kq;q)_{y_k}}\prod_{k=1}^n\frac{(bz_k,cz_k;q)_{y_k}}{(az_kq/b,az_kq/c;q)_{y_k}}\left(\frac{at_1\cdots t_nq}{bc}\right)^{y_k}\\
&=\prod_{1\leq j,k\leq n}\frac{(az_jz_kq;q)_{\infty}}{(az_jt_kz_kq;q)_{\infty}}\prod_{1\leq j< k\leq n}\frac{(at_jz_jt_kz_kq;q)_{\infty}}{(az_jz_kq;q)_{\infty}}\frac{(aq/bc;q)_{\infty}}{(at_1\cdots t_nq/bc;q)_{\infty}}\prod_{k=1}^n\frac{(at_kz_kq/b,at_kz_kq/c;q)_{\infty}}{(az_kq/b,az_kq/c;q)_{\infty}}
\end{align}
が$t_i=q^{m_i}$のおいて成り立つことを意味している. この右辺は各$t_i$について$t_i=0$で正則であり, 左辺も
\begin{align}
&\prod_{1\leq j,k\leq n}(z_k/t_jz_j;q)_{y_k}\prod_{k=1}^n\left(t_1\cdots t_n\right)^{y_k}\\
&=\prod_{1\leq j,k\leq n}t_j^{y_k}(z_k/t_jz_j;q)_{y_k}\\
&=\prod_{1\leq j,k\leq n}\prod_{i=0}^{y_k-1}(t_j-z_kq^i/z_j)
\end{align}
と書き換えられることから, 各$t_i$に関して正則である. よって, $t_i=q^{m_i}$における一致の定理を各$t_i$に関して用いることによって, 一般の$t_i$で
\begin{align}
&\sum_{0\leq y_1,\dots,y_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-az_jz_kq^{y_j+y_k}}{1-az_jz_k}\\
&\qquad\cdot\prod_{1\leq j,k\leq n}\frac{(az_jz_k,z_k/t_jz_j;q)_{y_k}}{(z_kq/z_j,at_jz_jz_kq;q)_{y_k}}\prod_{k=1}^n\frac{(bz_k,cz_k;q)_{y_k}}{(az_kq/b,az_kq/c;q)_{y_k}}\left(\frac{at_1\cdots t_nq}{bc}\right)^{y_k}\\
&=\prod_{1\leq j,k\leq n}\frac{(az_jz_kq;q)_{\infty}}{(az_jt_kz_kq;q)_{\infty}}\prod_{1\leq j< k\leq n}\frac{(at_jz_jt_kz_kq;q)_{\infty}}{(az_jz_kq;q)_{\infty}}\frac{(aq/bc;q)_{\infty}}{(at_1\cdots t_nq/bc;q)_{\infty}}\prod_{k=1}^n\frac{(at_kz_kq/b,at_kz_kq/c;q)_{\infty}}{(az_kq/b,az_kq/c;q)_{\infty}}
\end{align}
が成り立つことが分かる. ここで, $a=1$として$1/t_jz_j$を改めて$z_{n+j}$とすると,
\begin{align}
&\sum_{0\leq y_1,\dots,y_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k}}{1-z_jz_k}\\
&\qquad\cdot\prod_{1\leq j,k\leq n}\frac{(z_jz_k,z_{n+j}z_k;q)_{y_k}}{(z_kq/z_j,z_kq/z_{n+j};q)_{y_k}}\prod_{k=1}^n\frac{(bz_k,cz_k;q)_{y_k}}{(z_kq/b,z_kq/c;q)_{y_k}}\left(\frac{q}{bcz_1\cdots z_{2n}}\right)^{y_k}\\
&=\prod_{1\leq j,k\leq n}\frac{(z_jz_kq;q)_{\infty}}{(z_jq/z_{n+k};q)_{\infty}}\prod_{1\leq j< k\leq n}\frac{(q/z_{n+j}z_{n+k};q)_{\infty}}{(z_jz_kq;q)_{\infty}}\frac{(q/bc;q)_{\infty}}{(q/bcz_1\cdots z_{2n};q)_{\infty}}\prod_{k=1}^n\frac{(q/bz_{n+k},q/cz_{n+k};q)_{\infty}}{(z_kq/b,z_kq/c;q)_{\infty}}
\end{align}
となる. $b,c$を$z_{2n+1},z_{2n+2}$と置き換えると, 以下を得る.
\begin{align} &\sum_{0\leq y_1,\dots,y_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k}}{1-z_jz_k}\\ &\qquad\cdot\left(\frac{q}{z_1\cdots z_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_k;q)_{y_k}}{(z_kq/z_j;q)_{y_k}}\\ &=\frac{\prod_{1\leq j\leq k\leq n}(z_jz_kq;q)_{\infty}}{\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}(z_kq/z_j;q)_{\infty}}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q/z_jz_k;q)_{\infty}}{(q/z_1\cdots z_{2n+2};q)_{\infty}} \end{align}
$C_n$型Baileyの和公式
定理2と一致の定理を用いることにより, 以下を示すことができる.
\begin{align}
&\sum_{y_1,\dots,y_n\in\ZZ}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k}}{1-z_jz_k}\\
&\qquad\cdot \left(\frac{q}{a_1\cdots a_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(a_jz_k;q)_{y_k}}{(z_kq/a_j;q)_{y_k}}\\
&=\frac{\prod_{1\leq j\leq k\leq n}(z_jz_kq,q/z_jz_k;q)_{\infty}\prod_{1\leq j,k\leq n}(z_jq/z_k;q)_{\infty}}{\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}(z_kq/a_j,q/a_jz_k;q)_{\infty}}\frac{\prod_{1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}
\end{align}
が成り立つ.
$1\leq i\leq 2n+2$に対して, $t_i=1/a_i$とすると, 両辺は各$t_i$について$t_i=0$において正則であるから, 一致の定理より$m_i$を非負整数として, $t_i=q^{m_i}/z_i$, つまり$a_i=z_iq^{-m_i}$として示せば十分である. そのとき, 左辺は
\begin{align}
&\sum_{y_1,\dots,y_n\in\ZZ}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k}}{1-z_jz_k}\\
&\qquad\cdot \left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j};q)_{y_k}}{(z_kq^{m_j+1}/z_j;q)_{y_k}}\\
&=\left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j};q)_{-m_k}}{(z_kq^{m_j+1}/z_j;q)_{-m_k}}\\
&\qquad\cdot \prod_{1\leq j< k\leq n}\frac{z_jq^{-m_j}-z_kq^{-m_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{-m_j-m_k}}{1-z_jz_k}\\
&\qquad\cdot\sum_{0\leq y_1,\dots,y_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j-m_j}-z_kq^{y_k-m_k}}{z_jq^{-m_j}-z_kq^{-m_k}}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k-m_j-m_k}}{1-z_jz_kq^{-m_j-m_k}}\\
&\qquad\cdot \left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j-m_k};q)_{y_k}}{(z_kq^{m_j-m_k+1}/z_j;q)_{y_k}}\qquad y_i\mapsto y_i-m_i
\end{align}
となる ここで, 定理2より
\begin{align}
&\sum_{0\leq y_1,\dots,y_n}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j-m_j}-z_kq^{y_k-m_k}}{z_jq^{-m_j}-z_kq^{-m_k}}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k-m_j-m_k}}{1-z_jz_kq^{-m_j-m_k}}\\
&\qquad\cdot \left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j-m_k};q)_{y_k}}{(z_kq^{m_j-m_k+1}/z_j;q)_{y_k}}\\
&=\frac{\prod_{1\leq j\leq k\leq n}(z_jz_kq^{1-m_j-m_k};q)_{\infty}}{\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}(z_kq^{m_j-m_k+1}/z_j;q)_{\infty}}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q^{m_j+m_k+1}/z_jz_k;q)_{\infty}}{(q^{m_1+\cdots+m_{2n+2}+1}/z_1\cdots z_{2n+2};q)_{\infty}}
\end{align}
であるから, これを代入して,
\begin{align}
&\sum_{y_1,\dots,y_n\in\ZZ}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k}}{1-z_jz_k}\\
&\qquad\cdot \left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j};q)_{y_k}}{(z_kq^{m_j+1}/z_j;q)_{y_k}}\\
&=\left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j};q)_{-m_k}}{(z_kq^{m_j+1}/z_j;q)_{-m_k}}\\
&\qquad\cdot \prod_{1\leq j< k\leq n}\frac{z_jq^{-m_j}-z_kq^{-m_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{-m_j-m_k}}{1-z_jz_k}\\
&\qquad\cdot\frac{\prod_{1\leq j\leq k\leq n}(z_jz_kq^{1-m_j-m_k};q)_{\infty}}{\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}(z_kq^{m_j-m_k+1}/z_j;q)_{\infty}}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q^{m_j+m_k+1}/z_jz_k;q)_{\infty}}{(q^{m_1+\cdots+m_{2n+2}+1}/z_1\cdots z_{2n+2};q)_{\infty}}\\
&=\left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_k;q)_{-m_j-m_k}(z_kq/z_j;q)_{m_j}}{(z_jz_k;q)_{-m_j}(z_kq/z_j;q)_{m_j-m_k}}\\
&\qquad\cdot\prod_{1\leq j\leq k\leq n}\frac{(z_jz_kq;q)_{\infty}}{(z_jz_kq;q)_{-m_j-m_k}}\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}\frac {(z_kq/z_j;q)_{m_j-m_k}}{(z_kq/z_j;q)_{\infty}}\\
&\qquad\cdot \prod_{1\leq j< k\leq n}\frac{z_jq^{-m_j}-z_kq^{-m_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{-m_j-m_k}}{1-z_jz_k}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}\\
&=\left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(-z_jz_k)^{-m_k}q^{\binom{m_j+m_k+1}2-\binom{m_j+1}2}(q/z_jz_k,z_kq/z_j;q)_{m_j}}{(q/z_jz_k;q)_{m_j+m_k}}\prod_{1\leq j,k\leq n}\frac 1{(z_kq/z_j;q)_{m_j-m_k}}\\
&\qquad\cdot\prod_{1\leq j\leq k\leq n}(z_jz_kq;q)_{\infty}(1/z_jz_k;q)_{m_j+m_k}(-z_jz_kq)^{m_j+m_k}q^{-\binom{m_j+m_k+1}2}\\
&\qquad\cdot\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}\frac {1}{(z_kq/z_j;q)_{\infty}}\prod_{1\leq j< k\leq n}\frac{z_jq^{-m_j}-z_kq^{-m_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{-m_j-m_k}}{1-z_jz_k}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}\\
&=\left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_k)^{-m_k}q^{m_jm_k+\binom{m_k+1}2}(q/z_jz_k,z_kq/z_j;q)_{m_j}}{(q/z_jz_k;q)_{m_j+m_k}}\\
&\qquad\cdot\prod_{1\leq j< k\leq n}\frac 1{(z_kq/z_j;q)_{m_j-m_k}(z_jq/z_k;q)_{m_k-m_j}}\\
&\qquad\cdot\prod_{1\leq j\leq k\leq n}(z_jz_kq;q)_{\infty}(1/z_jz_k;q)_{m_j+m_k}(-z_jz_k)^{m_j+m_k}q^{-\binom{m_j}2-m_jm_k-\binom{m_k}2}\\
&\qquad\cdot\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}\frac {1}{(z_kq/z_j;q)_{\infty}}\prod_{1\leq j< k\leq n}\frac{z_jq^{-m_j}-z_kq^{-m_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{-m_j-m_k}}{1-z_jz_k}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}\\
&=\left(\frac{q}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}(z_jz_k)^{-m_k}q^{\binom{m_k+1}2}(q/z_jz_k,z_kq/z_j;q)_{m_j}\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}\frac 1{(q/z_jz_k;q)_{m_j+m_k}}\prod_{1\leq j< k\leq n}\frac 1{(q/z_jz_k;q)_{m_j+m_k}}\\
&\qquad\cdot\prod_{1\leq j< k\leq n}\frac{1-z_j/z_k}{1-z_jq^{m_k-m_j}/z_k}(-z_kq/z_j)^{m_k-m_j}q^{-\binom{m_j-m_k}2}\\
&\qquad\cdot\prod_{1\leq j\leq k\leq n}\frac{(z_jz_kq;q)_{\infty}(1/z_jz_k;q)_{m_j+m_k}(-z_jz_k)^{m_j+m_k}q^{-\binom{m_j}2-m_jm_k-\binom{m_k}2}}{(q/z_jz_k;q)_{m_j+m_k}}\\
&\qquad\cdot\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}\frac {1}{(z_kq/z_j;q)_{\infty}}\prod_{1\leq j< k\leq n}\frac{z_jq^{-m_j}-z_kq^{-m_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{-m_j-m_k}}{1-z_jz_k}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}\\
&=\left(\frac{q}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}(z_jz_k)^{-m_k}q^{\binom{m_k+1}2}\frac{(q/z_jz_k,z_kq/z_j;q)_{\infty}}{(q/a_jz_k,z_kq/a_j;q)_{\infty}}\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}\frac {(q/a_ja_k;q)_{\infty}}{(q/z_jz_k;q)_{\infty}}\prod_{1\leq j< k\leq n}\frac{(q/a_ja_k;q)_{\infty}}{(q/z_jz_k;q)_{\infty}}\\
&\qquad\cdot\prod_{1\leq j< k\leq n}q^{-m_k}(-z_kq/z_j)^{m_k-m_j}q^{-\binom{m_j-m_k}2}\\
&\qquad\cdot\prod_{1\leq j\leq k\leq n}(z_jz_kq;q)_{\infty}(-z_jz_k)^{m_j+m_k}q^{-\binom{m_j+1}2-m_jm_k-\binom{m_k+1}2}\\
&\qquad\cdot\prod_{\substack{n+1\leq j\leq 2n+2\\1\leq k\leq n}}\frac {1}{(z_kq/z_j;q)_{\infty}}\frac{\prod_{n+1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}
\end{align}
となる. まとめると,
\begin{align}
&\sum_{y_1,\dots,y_n\in\ZZ}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k}}{1-z_jz_k}\\
&\qquad\cdot \left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j};q)_{y_k}}{(z_kq^{m_j+1}/z_j;q)_{y_k}}\\
&=\left(\frac{q}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}(z_jz_k)^{-m_k}q^{\binom{m_k+1}2}\\
&\qquad\cdot\prod_{1\leq j\leq k\leq n}(-z_jz_k)^{m_j+m_k}q^{-\binom{m_j+1}2-m_jm_k-\binom{m_k+1}2}\\
&\qquad\cdot\prod_{1\leq j< k\leq n}q^{-m_k}(-z_kq/z_j)^{m_k-m_j}q^{-\binom{m_j-m_k}2}\\
&\qquad\cdot \frac{\prod_{1\leq j\leq k\leq n}(z_jz_kq,q/z_jz_k;q)_{\infty}\prod_{1\leq j,k\leq n}(z_kq/z_j;q)_{\infty}}{\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}(q/a_jz_k,z_kq/a_j;q)_{\infty}}\frac{\prod_{1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}
\end{align}
となる. ここで,
\begin{align}
&\left(\frac{q}{z_1\cdots z_{2n+2}}\right)^{-m_1-\cdots-m_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}(z_jz_k)^{-m_k}q^{\binom{m_k+1}2}\\
&\qquad\cdot\prod_{1\leq j\leq k\leq n}(-z_jz_k)^{m_j+m_k}q^{-\binom{m_j+1}2-m_jm_k-\binom{m_k+1}2}\\
&\qquad\cdot\prod_{1\leq j< k\leq n}q^{-m_k}(-z_kq/z_j)^{m_k-m_j}q^{-\binom{m_j-m_k}2}\\
&=q^{-m_1-\cdots-m_n}\left(\prod_{\substack{1\leq k\leq n}}z_k^{-m_k}q^{\binom{m_k+1}2}\right)^{2n+2}\\
&\qquad\cdot\prod_{1\leq j\leq n}z_j^{4m_j}q^{-2m_j^2-m_j}\\
&\qquad\cdot\prod_{1\leq j< k\leq n}(-z_jz_k)^{m_j+m_k}(-z_k/z_j)^{m_k-m_j}q^{-\binom{m_j+1}2-m_jm_k-\binom{m_k+1}2-m_j-\binom{m_j-m_k}2}\\
&=\left(\prod_{\substack{1\leq k\leq n}}z_k^{-m_k}q^{\binom{m_k+1}2}\right)^{2n+2}\prod_{1\leq j\leq n}z_j^{4m_j}q^{-2m_j^2-2m_j}\prod_{1\leq j< k\leq n}z_j^{2m_j}z_k^{2m_k}q^{-m_j^2-m_j-m_k^2-m_k}\\
&=\left(\prod_{\substack{1\leq k\leq n}}z_k^{-m_k}q^{\binom{m_k+1}2}\right)^{2n+2}\prod_{1\leq j\leq n}z_j^{2m_j}q^{-m_j^2-m_j}\prod_{1\leq j,k\leq n}z_j^{2m_j}q^{-m_j^2-m_j}\\
&=1
\end{align}
となるから,
\begin{align}
&\sum_{y_1,\dots,y_n\in\ZZ}\prod_{1\leq j< k\leq n}\frac{z_jq^{y_j}-z_kq^{y_k}}{z_j-z_k}\prod_{1\leq j\leq k\leq n}\frac{1-z_jz_kq^{y_j+y_k}}{1-z_jz_k}\\
&\qquad\cdot \left(\frac{q^{m_1+\cdots+m_{2n+2}+1}}{z_1\cdots z_{2n+2}}\right)^{y_1+\cdots+y_n}\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}\frac{(z_jz_kq^{-m_j};q)_{y_k}}{(z_kq^{m_j+1}/z_j;q)_{y_k}}\\
&=\frac{\prod_{1\leq j\leq k\leq n}(z_jz_kq,q/z_jz_k;q)_{\infty}\prod_{1\leq j,k\leq n}(z_kq/z_j;q)_{\infty}}{\prod_{\substack{1\leq j\leq 2n+2\\1\leq k\leq n}}(q/a_jz_k,z_kq/a_j;q)_{\infty}}\frac{\prod_{1\leq j< k\leq 2n+2}(q/a_ja_k;q)_{\infty}}{(q/a_1\cdots a_{2n+2};q)_{\infty}}
\end{align}
となって示すべきことが得られる.
