RosengrenによるC_n型Jacksonの和公式

$a^2q^{3-n}=bcde$として, WarnaarによるC_n型Jacksonの和公式 において, $N=1$とした式
\begin{align} &\sum_{y_1,\dots,y_n=0}^1\prod_{1\leq i< j\leq n}\frac{\theta(z_iq^{y_i-y_j}/z_j;p)q^{y_j}}{\theta(z_i/z_j;p)}\frac{\theta(az_iz_jq^{y_i+y_j};p)}{\theta(az_iz_jq;p)}\\ &\qquad\cdot\prod_{i=1}^n\frac{\theta(az_i^2q^{2y_i};p)(az_i^2,bz_i,cz_i,dz_i,ez_i,q^{-1};q,p)_{y_i}q^{y_i}}{\theta(az_i^2;p)(q,az_iq/b,az_iq/c,az_iq/d,az_iq/e,az_i^2q^2;q,p)_{y_i}}\\ &=\prod_{i=1}^n\frac{\theta(az_i^2q,aq^{2-i}/bc,aq^{2-i}/bd,aq^{2-i}/cd;p)}{\theta(aq^{2-n}/bcdz_i,az_iq/b,az_iq/c,az_iq/d;p)} \end{align}
から始める. $y_i\in\{0,1\}$より
\begin{align} \frac{(q^{-1};q,p)_{y_i}}{(q;q,p)_{y_i}}q^{y_i}&=(-1)^{y_i}\\ \frac{\theta(az_i^2q^{2y_i};p)}{\theta(az_i^2;p)}\frac{(az_i^2;q,p)_{y_i}}{(az_i^2q^2;q,p)_{y_i}}&=1 \end{align}
であるから, 上の式は
\begin{align} &\sum_{y_1,\dots,y_n=0}^1\prod_{1\leq i< j\leq n}\frac{\theta(z_iq^{y_i-y_j}/z_j;p)q^{y_j}}{\theta(z_i/z_j;p)}\frac{\theta(az_iz_jq^{y_i+y_j};p)}{\theta(az_iz_j;p)}\\ &\qquad\cdot\prod_{i=1}^n\frac{(bz_i,cz_i,dz_i,ez_i;q,p)_{y_i}(-1)^{y_i}}{(az_iq/b,az_iq/c,az_iq/d,az_iq/e;q,p)_{y_i}}\\ &=\prod_{1\leq i< j\leq n}\frac{\theta(az_iz_jq;p)}{\theta(az_iz_j;p)}\prod_{i=1}^n\frac{\theta(az_i^2q,aq^{2-i}/bc,aq^{2-i}/bd,aq^{2-i}/cd;p)}{\theta(az_iq/b,az_iq/c,az_iq/d,eq^{-1}/az_i;p)} \end{align}
と書き換えられる. ここで, $q\mapsto q^{-1}$としてから$a\mapsto aq^2$とすると($y_i\in\{0,1\}$により現れる上昇冪は$q$によらない),
\begin{align} &\sum_{y_1,\dots,y_n=0}^1\prod_{1\leq i< j\leq n}\frac{\theta(z_iq^{y_j-y_i}/z_j;p)q^{-y_j}}{\theta(z_i/z_j;p)}\frac{\theta(az_iz_jq^{2-y_i-y_j};p)}{\theta(az_iz_jq^2;p)}\\ &\qquad\cdot\prod_{i=1}^n\frac{(bz_i,cz_i,dz_i,ez_i;q,p)_{y_i}(-1)^{y_i}}{(az_iq/b,az_iq/c,az_iq/d,az_iq/e;q,p)_{y_i}}\\ &=\prod_{1\leq i< j\leq n}\frac{\theta(az_iz_jq;p)}{\theta(az_iz_jq^2;p)}\prod_{i=1}^n\frac{\theta(az_i^2q,aq^i/bc,aq^i/bd,aq^i/cd;p)}{\theta(az_iq/b,az_iq/c,az_iq/d,eq^{-1}/az_i;p)} \end{align}
となる. ここで, 条件は$a^2q^{n+1}=bcde$に書き換わる. ここで, $m:=m_1+\cdots+m_n$として, $z_1,\dots,z_n$を$w_1,\dots,w_{m}$に付け替えた式に
\begin{align} &\sum_{y_1,\dots,y_m=0}^1\prod_{1\leq i< j\leq m}\frac{\theta(w_iq^{y_j-y_i}/w_j;p)q^{-y_j}}{\theta(w_i/w_j;p)}\frac{\theta(aw_iw_jq^{2-y_i-y_j};p)}{\theta(aw_iw_jq^2;p)}\\ &\qquad\cdot\prod_{i=1}^m\frac{(bw_i,cw_i,dw_i,ew_i;q,p)_{y_i}(-1)^{y_i}}{(aw_iq/b,aw_iq/c,aw_iq/d,aw_iq/e;q,p)_{y_i}}\\ &=\prod_{1\leq i< j\leq m}\frac{\theta(aw_iw_jq;p)}{\theta(aw_iw_jq^2;p)}\prod_{i=1}^m\frac{\theta(aw_i^2q,aq^i/bc,aq^i/bd,aq^i/cd;p)}{\theta(aw_iq/b,aw_iq/c,aw_iq/d,eq^{-1}/aw_i;p)} \end{align}
において,
\begin{align} (w_1,\dots,w_m)=(z_1,z_1q,\dots,z_1q^{m_1-1},\dots,z_n,z_nq,\dots,z_nq^{m_n-1}) \end{align}
とすると,
\begin{align} (y_1,\dots,y_m)=(\underbrace{\overbrace{1,\dots,1}^{x_1},0,\dots,0}_{m_1},\dots,\underbrace{\overbrace{1,\dots,1}^{x_n},0,\dots,0}_{m_n}) \end{align}
の形の元以外では
\begin{align} \prod_{1\leq i< j\leq m}\frac{\theta(w_iq^{y_j-y_i}/w_j;p)q^{-y_j}}{\theta(w_i/w_j;p)} \end{align}
が$0$になる. このような場合,
\begin{align} \prod_{k=1}^{m}(bw_k;q,p)_{y_k}&=\prod_{k=1}^n(bz_k;q,p)_{x_k}\\ \prod_{k=1}^{m}\theta(bw_k;p)&=\prod_{k=1}^n(bz_k;q,p)_{m_k} \end{align}
と書き換えられる. また, 二重の積については, $f(x,y)=\pm f(y,x)$であるとき,
\begin{align} &\prod_{1\leq i< j\leq m}\frac{f(w_iq^{-y_i},w_jq^{-y_j})}{f(w_i,w_j)}\\ &=\prod_{1\leq i,j\leq n}\prod_{t=x_i+1}^{m_i}\prod_{u=1}^{x_j}\frac{f(z_iq^{t-1},z_jq^{u-2})}{f(z_iq^{t-1},z_jq^{u-1})}\prod_{i=1}^n\prod_{1\leq t< u\leq x_i}\frac{f(z_iq^{t-2},z_iq^{u-2})}{f(z_iq^{t-1},z_iq^{u-1})}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\prod_{t=1}^{x_i}\prod_{u=1}^{x_j}\frac{f(z_iq^{t-2},z_jq^{u-2})}{f(z_iq^{t-1},z_jq^{u-1})} \end{align}
と書き換えられる. 特に, $f(x,y)=\theta(axyq^2;p)$として,
\begin{align} &\prod_{1\leq i< j\leq m}\frac{\theta(aw_iw_jq^{2-y_i-y_j};p)}{\theta(aw_iw_jq^2;p)}\\ &=\prod_{1\leq i,j\leq n}\prod_{t=x_i+1}^{m_i}\prod_{u=1}^{x_j}\frac{\theta(az_iz_jq^{t+u-1};p)}{\theta(az_iz_jq^{t+u};p)}\prod_{i=1}^n\prod_{1\leq t< u\leq x_i}\frac{\theta(az_i^2q^{t+u-2};p)}{\theta(az_i^2q^{t+u};p)}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\prod_{t=1}^{x_i}\prod_{u=1}^{x_j}\frac{\theta(az_iz_jq^{t+u-2};p)}{\theta(az_iz_jq^{t+u};p)}\\ &=\prod_{1\leq i,j\leq n}\prod_{u=1}^{x_j}\frac{\theta(az_iz_jq^{x_i+u};p)}{\theta(az_iz_jq^{m_i+u};p)}\prod_{i=1}^n\prod_{u=1}^{x_i}\frac{\theta(az_i^2q^{u-1},az_i^2q^{u};p)}{\theta(az_i^2q^{2u-2},az_i^2q^{2u-1};p)}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\prod_{u=1}^{x_j}\frac{\theta(az_iz_jq^{u-1},az_iz_jq^{u};p)}{\theta(az_iz_jq^{x_i+u-1},az_iz_jq^{x_i+u};p)}\\ &=\prod_{1\leq i,j\leq n}\frac{(az_iz_jq^{x_i+1};q,p)_{x_j}}{(az_iz_jq^{m_i+1};q,p)_{x_j}}\prod_{i=1}^n\frac{(az_i^2q;q,p)_{x_i}}{(az_i^2q^{x_i};q,p)_{x_i}}\prod_{1\leq i< j\leq n}\frac{(az_iz_j,az_iz_jq;q,p)_{x_j}}{(az_iz_jq^{x_i},az_iz_jq^{x_i+1};q,p)_{x_j}}\\ &=\prod_{1\leq i,j\leq n}\frac{1}{(az_iz_jq^{m_i+1};q,p)_{x_j}}\prod_{i=1}^n\frac{(az_i^2q;q,p)_{x_i}(az_i^2q^{x_i+1};q,p)_{x_i}}{(az_i^2q^{x_i};q,p)_{x_i}}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\frac{(az_iz_jq^{x_j+1};q,p)_{x_i}(az_iz_j,az_iz_jq;q,p)_{x_j}}{(az_iz_jq^{x_i};q,p)_{x_j}}\\ &=\prod_{1\leq i,j\leq n}\frac{1}{(az_iz_jq^{m_i+1};q,p)_{x_j}}\prod_{i=1}^n\frac{(az_i^2q;q,p)_{x_i}(az_i^2q^{x_i+1};q,p)_{x_i}}{(az_i^2q^{x_i};q,p)_{x_i}}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\frac{(az_iz_jq;q,p)_{x_i+x_j}(az_iz_j;q,p)_{x_i}(az_iz_j;q,p)_{x_j}}{(az_iz_j;q,p)_{x_i+x_j}}\\ &=\prod_{1\leq i,j\leq n}\frac{(az_iz_j;q,p)_{x_j}}{(az_iz_jq^{m_i+1};q,p)_{x_j}}\prod_{1\leq i\leq j\leq n}\frac{\theta(az_iz_jq^{x_i+x_j};p)}{\theta(az_iz_j;p)} \end{align}
を得る. つまり,
\begin{align} \prod_{1\leq i< j\leq m}\frac{\theta(aw_iw_jq^{2-y_i-y_j};p)}{\theta(aw_iw_jq^2;p)}&=\prod_{1\leq i,j\leq n}\frac{(az_iz_j;q,p)_{x_j}}{(az_iz_jq^{m_i+1};q,p)_{x_j}}\prod_{1\leq i\leq j\leq n}\frac{\theta(az_iz_jq^{x_i+x_j};p)}{\theta(az_iz_j;p)} \end{align}
が得られた. 次に, $f(x,y)=\theta(x/y;p)y$として,
\begin{align} &\prod_{1\leq i< j\leq m}\frac{\theta(w_iq^{y_j-y_i}/w_j;p)q^{-y_j}}{\theta(w_i/w_j;p)}\\ &=\prod_{1\leq i,j\leq n}\prod_{t=x_i+1}^{m_i}\prod_{u=1}^{x_j}\frac{\theta(z_iq^{t-u+1}/z_j;p)q^{-1}}{\theta(z_iq^{t-u}/z_j;p)}\prod_{i=1}^n\prod_{1\leq t< u\leq x_i}q^{-1}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\prod_{t=1}^{x_i}\prod_{u=1}^{x_j}q^{-1}\\ &=\prod_{1\leq i,j\leq n}\prod_{u=1}^{x_j}\frac{\theta(z_iq^{m_i-u+1}/z_j;p)q^{x_i-m_i}}{\theta(z_iq^{x_i-u+1}/z_j;p)}\prod_{i=1}^nq^{-\binom{x_i}2}\prod_{1\leq i< j\leq n}q^{-x_ix_j}\\ &=\prod_{1\leq i,j\leq n}\frac{(z_iq^{m_i-x_j+1}/z_j;q,p)_{x_j}q^{x_ix_j-m_ix_j}}{(z_iq^{x_i-x_j+1}/z_j;q,p)_{x_j}}\prod_{i=1}^nq^{-\binom{x_i}2}\prod_{1\leq i< j\leq n}q^{-x_ix_j}\\ &=\prod_{1\leq i,j\leq n}(z_jq^{-m_i}/z_i;q,p)_{x_j}(-z_iq^{x_i}/z_j)^{x_j}q^{-\binom{x_j}2}\prod_{i=1}^n\frac{q^{-\binom{x_i}2}}{(q;q,p)_{x_i}}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\frac{q^{-x_ix_j}}{(z_iq^{x_i-x_j+1}/z_j;q,p)_{x_j}(z_jq^{x_j-x_i+1}/z_i;q,p)_{x_i}}\\ &=\prod_{1\leq i,j\leq n}(z_jq^{-m_i}/z_i;q,p)_{x_j}(-z_iq^{x_i}/z_j)^{x_j}q^{-\binom{x_j}2}\prod_{i=1}^n\frac{q^{-\binom{x_i}2}}{(q;q,p)_{x_i}}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}\frac{q^{-x_ix_j}(z_iq/z_j;q,p)_{x_i-x_j}(z_jq/z_i;q,p)_{x_j-x_i}}{(z_iq/z_j;q,p)_{x_i}(z_jq/z_i;q,p)_{x_j}}\\ &=(-q)^{x_1+\cdots+x_n}\prod_{1\leq i,j\leq n}\frac{(z_jq^{-m_i}/z_i;q,p)_{x_j}}{(z_jq/z_i;q,p)_{x_j}}\\ &\qquad\cdot \prod_{1\leq i< j\leq n}(-z_iq^{x_i}/z_j)^{x_j}q^{-\binom{x_j}2}(-z_jq^{x_j}/z_i)^{x_i}q^{-\binom{x_i}2}\\ &\qquad\cdot\prod_{1\leq i< j\leq n}q^{-x_ix_j}\frac{(z_jq/z_i;q,p)_{x_j-x_i}}{(z_j/z_i;q,p)_{x_j-x_i}}(-z_iq/z_j)^{x_i-x_j}q^{\binom{x_i-x_j}2}\\ &=(-q)^{x_1+\cdots+x_n}\prod_{1\leq i,j\leq n}\frac{(z_jq^{-m_i}/z_i;q,p)_{x_j}}{(z_jq/z_i;q,p)_{x_j}}\prod_{1\leq i< j\leq n}\frac{\theta(z_jq^{x_j-x_i}/z_i;p)q^{x_i}}{\theta(z_j/z_i;p)} \end{align}
つまり,
\begin{align} &\prod_{1\leq i< j\leq m}\frac{\theta(w_iq^{y_j-y_i}/w_j;p)q^{-y_j}}{\theta(w_i/w_j;p)}\\ &=(-q)^{x_1+\cdots+x_n}\prod_{1\leq i,j\leq n}\frac{(z_jq^{-m_i}/z_i;q,p)_{x_j}}{(z_jq/z_i;q,p)_{x_j}}\prod_{1\leq i< j\leq n}\frac{\theta(z_iq^{x_i-x_j}/z_j;p)q^{x_j}}{\theta(z_i/z_j;p)} \end{align}
を得る. また,
\begin{align} &\prod_{1\leq i< j\leq m}\frac{\theta(aw_iw_jq;p)}{\theta(aw_iw_jq^2;p)}\prod_{i=1}^m\theta(aw_i^2q;p)\\ &=\prod_{i=1}^n\prod_{1\leq t< u\leq m_i}\frac{\theta(az_i^2q^{t+u-1};p)}{\theta(az_i^2q^{t+u};p)}\left(\prod_{1\leq i< j\leq n}\prod_{t=1}^{m_i}\prod_{u=1}^{m_j}\frac{\theta(az_iz_jq^{t+u-1};p)}{\theta(az_iz_jq^{t+u};p)}\right)\prod_{i=1}^n\prod_{u=1}^{m_i}\theta(az_i^2q^{2u-1};p)\\ &=\prod_{i=1}^n\prod_{1\leq u\leq m_i}\theta(az_i^2q^{u};p)\prod_{1\leq i< j\leq n}\prod_{u=1}^{m_j}\frac{\theta(az_iz_jq^{u};p)}{\theta(az_iz_jq^{m_i+u};p)}\\ &=\prod_{i=1}^n(az_i^2q;q,p)_{m_i}\prod_{1\leq i< j\leq n}\frac{(az_iz_jq;q,p)_{m_i}(az_iz_jq;q,p)_{m_j}}{(az_iz_jq;q,p)_{m_i+m_j}}\\ &=\prod_{1\leq i,j\leq n}(az_iz_jq;q,p)_{m_i}\prod_{1\leq i< j\leq n}\frac{1}{(az_iz_jq;q,p)_{m_i+m_j}} \end{align}
である. これらを代入すると,
\begin{align} &\sum_{y_1,\dots,y_m=0}^1\prod_{1\leq i< j\leq m}\frac{\theta(w_iq^{y_j-y_i}/w_j;p)q^{-y_j}}{\theta(w_i/w_j;p)}\frac{\theta(aw_iw_jq^{2-y_i-y_j};p)}{\theta(aw_iw_jq^2;p)}\\ &\qquad\cdot\prod_{i=1}^m\frac{(bw_i,cw_i,dw_i,ew_i;q,p)_{y_i}(-1)^{y_i}}{(aw_iq/b,aw_iq/c,aw_iq/d,aw_iq/e;q,p)_{y_i}}\\ &=\prod_{1\leq i< j\leq m}\frac{\theta(aw_iw_jq;p)}{\theta(aw_iw_jq^2;p)}\prod_{i=1}^m\frac{\theta(aw_i^2q,aq^i/bc,aq^i/bd,aq^i/cd;p)}{\theta(aw_iq/b,aw_iq/c,aw_iq/d,eq^{-1}/aw_i;p)} \end{align}
は以下のように書き換えられ, 条件は$aq^{m+1}=bcde$となって定理1を得る.
\begin{align} &\sum_{x_1,\dots,x_n=0}^{m_1,\dots,m_n}\prod_{1\leq i< j\leq n}\frac{\theta(z_iq^{x_i-x_j}/z_j;p)q^{x_j}}{\theta(z_i/z_j;p)}\prod_{1\leq i\leq j\leq n}\frac{\theta(az_iz_jq^{x_i+x_j};p)}{\theta(az_iz_j;p)}\\ &\qquad\cdot \prod_{1\leq i,j\leq n}\frac{(z_jq^{-m_i}/z_i,az_iz_j;q,p)_{x_j}}{(z_jq/z_i,az_iz_jq^{m_i+1};q,p)_{x_j}}\\ &\qquad\cdot\prod_{i=1}^n\frac{(bz_i,cz_i,dz_i,ez_i;q,p)_{x_i}q^{x_i}}{(az_iq/b,az_iq/c,az_iq/d,az_iq/e;q,p)_{x_i}}\\ &=\frac{\prod_{1\leq i,j\leq n}(az_iz_jq;q,p)_{m_i}}{\prod_{1\leq i< j\leq n}(az_iz_jq;q,p)_{m_i+m_j}}\frac{(aq/bc,aq/bd,aq/cd;q,p)_m}{\prod_{i=1}^n(az_iq/b,az_iq/c,az_iq/d,eq^{-m_i}/az_i;q,p)_{m_i}} \end{align}