Kajihara-Noumiの変換公式
前の記事 の記法を用いる. 前の記事 の定理2の一般化として, 以下の変換公式が知られている.
$m,n\geq 1, w_1\cdots w_m=a_1\cdots a_{m+n}z_1\cdots z_n$であるとき,
\begin{align}
&\sum_{\substack{0\leq y_1,\dots,y_n\\y_1+\cdots+y_n=N}}\frac{\Delta(z_1q^{y_1},\dots,z_nq^{y_n};p)}{\Delta(z_1,\dots,z_n;p)}\prod_{k=1}^n\frac{(a_1z_k,\dots,a_{m+n}z_k;q,p)_{y_k}}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_n;q,p)_{y_k}}\\
&=\sum_{\substack{0\leq y_1,\dots,y_m\\y_1+\cdots+y_m=N}}\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m};p)}{\Delta(w_1,\dots,w_m;p)}\prod_{k=1}^m\frac{(w_k/a_1,\dots,w_k/a_{m+n};q,p)_{y_k}}{(w_kz_1,\dots,w_kz_n,w_kq/w_1,\dots,w_kq/w_m;q,p)_{y_k}}
\end{align}
が成り立つ.
以下はRosengrenによる2006年の論文における証明である.
$n$に関する帰納法を用いる. $n=1$のときは
前の記事
の定理2から成り立つことが分かる. 固定した$n\geq 1$に対して定理が成り立つと仮定する. このとき,
\begin{align}
&\sum_{\substack{0\leq y_1,\dots,y_{n+1}\\y_1+\cdots+y_{n+1}=N}}\frac{\Delta(z_1q^{y_1},\dots,z_{n+1}q^{y_{n+1}};p)}{\Delta(z_1,\dots,z_{n+1};p)}\prod_{k=1}^{n+1}\frac{(a_1z_k,\dots,a_{m+n+1}z_k;q,p)_{y_k}}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_{n+1};q,p)_{y_k}}\\
&=\sum_{0\leq s}\frac{(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_s}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}q/z_1,\dots,z_{n+1}q/z_{n},q;q,p)_s}\\
&\qquad\cdot\sum_{\substack{0\leq y_1,\dots,y_{n}\\y_1+\cdots+y_{n}=N-s}}\frac{\Delta(z_1q^{y_1},\dots,z_nq^{y_n};p)}{\Delta(z_1,\dots,z_{n};p)}\prod_{j=1}^n\frac{z_jq^{y_j}\theta(z_{n+1}q^{s-y_j}/z_j;p)}{z_j\theta(z_{n+1}/z_j;p)}\\
&\qquad\cdot\prod_{k=1}^{n}\frac{(a_1z_k,\dots,a_{m+n+1}z_k;q,p)_{y_k}}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_{n+1};q,p)_{y_k}}\qquad(s=y_{n+1})\\
&=\sum_{0\leq s}\frac{(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_s}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}/z_1,\dots,z_{n+1}/z_{n},q;q,p)_s}\\
&\qquad\cdot\sum_{\substack{0\leq y_1,\dots,y_{n}\\y_1+\cdots+y_{n}=N-s}}\frac{\Delta(z_1q^{y_1},\dots,z_nq^{y_n};p)}{\Delta(z_1,\dots,z_{n};p)}\\
&\qquad\cdot\prod_{k=1}^{n}\frac{(a_1z_k,\dots,a_{m+n+1}z_k;q,p)_{y_k}\theta(z_kq^{y_k-s}/z_{n+1};p)}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_{n+1};q,p)_{y_k}\theta(z_kq^{-s}/z_{n+1};p)}
\end{align}
であり, 内側の和は$a_1',\dots,a_{m+n+2}'$を$a_1,\dots,a_{m+n+1},q^{1-s}/z_{n+1}$として, $w_1',\dots,w_{m+2}'$を$w_1,\dots,w_m,q/z_{n+1},q^{-s}/z_{n+1}$としたものに対して帰納法の仮定を用いると,
\begin{align}
&\sum_{\substack{0\leq y_1,\dots,y_{n}\\y_1+\cdots+y_{n}=N-s}}\frac{\Delta(z_1q^{y_1},\dots,z_nq^{y_n};p)}{\Delta(z_1,\dots,z_{n};p)}\\
&\qquad\cdot\prod_{k=1}^{n}\frac{(a_1z_k,\dots,a_{m+n+1}z_k;q,p)_{y_k}\theta(z_kq^{y_k-s}/z_{n+1};p)}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_{n+1};q,p)_{y_k}\theta(z_kq^{-s}/z_{n+1};p)}\\
&=\sum_{\substack{0\leq y_1,\dots,y_{n}\\y_1+\cdots+y_{n}=N-s}}\frac{\Delta(z_1q^{y_1},\dots,z_nq^{y_n};p)}{\Delta(z_1,\dots,z_{n};p)}\\
&\qquad\cdot\prod_{k=1}^{n}\frac{(a_1z_k,\dots,a_{m+n+1}z_k,z_kq^{1-s}/z_{n+1};q,p)_{y_k}}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_{n+1},z_kq^{-s}/z_{n+1};q,p)_{y_k}}\\
&=\sum_{\substack{0\leq y_1,\dots,y_m,t,u\\y_1+\cdots+y_m+t+u=N-s}}\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m},q^{t+1}/z_{n+1},q^{u-s}/z_{n+1};p)}{\Delta(w_1,\dots,w_m,q/z_{n+1},q^{-s}/z_{n+1};p)}\\
&\qquad\cdot\prod_{k=1}^m\frac{(w_k/a_1,\dots,w_k/a_{m+n+1},w_kz_{n+1}q^{s-1};q,p)_{y_k}}{(w_kz_1,\dots,w_kz_n,w_kq/w_1,\dots,w_kq/w_m,w_kz_{n+1},w_kz_{n+1}q^{s+1};q,p)_{y_k}}\\
&\qquad\cdot\frac{(q/a_1z_{n+1},\dots,q/a_{m+n+1}z_{n+1},q^{s};q,p)_t}{(z_1q/z_{n+1},\dots,z_nq/z_{n+1},q^2/w_1z_{n+1},\dots,q^2/w_nz_{n+1},q,q^{s+2};q,p)_t}\\
&\qquad\cdot\frac{(q^{-s}/a_1z_{n+1},\dots,q^{-s}/a_{m+n+1}z_{n+1},q^{-1};q,p)_u}{(z_1q^{-s}/z_{n+1},\dots,z_nq^{-s}/z_{n+1},q^{1-s}/w_1z_{n+1},\dots,q^{1-s}/w_nz_{n+1},q^{-s},q;q,p)_u}
\end{align}
となる. よって,
\begin{align}
&\sum_{\substack{0\leq y_1,\dots,y_{n+1}\\y_1+\cdots+y_{n+1}=N}}\frac{\Delta(z_1q^{y_1},\dots,z_{n+1}q^{y_{n+1}};p)}{\Delta(z_1,\dots,z_{n+1};p)}\prod_{k=1}^{n+1}\frac{(a_1z_k,\dots,a_{m+n+1}z_k;q,p)_{y_k}}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_{n+1};q,p)_{y_k}}\\
&=\sum_{0\leq s}\frac{(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_s}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}/z_1,\dots,z_{n+1}/z_{n},q;q,p)_s}\\
&\qquad\cdot\sum_{\substack{0\leq y_1,\dots,y_m,t,u\\y_1+\cdots+y_m+t+u=N-s}}\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m},q^{t+1}/z_{n+1},q^{u-s}/z_{n+1};p)}{\Delta(w_1,\dots,w_m,q/z_{n+1},q^{-s}/z_{n+1};p)}\\
&\qquad\cdot\prod_{k=1}^m\frac{(w_k/a_1,\dots,w_k/a_{m+n+1},w_kz_{n+1}q^{s-1};q,p)_{y_k}}{(w_kz_1,\dots,w_kz_{n+1},w_kq/w_1,\dots,w_kq/w_m,w_kz_{n+1}q^{s+1};q,p)_{y_k}}\\
&\qquad\cdot\frac{(q/a_1z_{n+1},\dots,q/a_{m+n+1}z_{n+1},q^{s};q,p)_t}{(z_1q/z_{n+1},\dots,z_nq/z_{n+1},q^2/w_1z_{n+1},\dots,q^2/w_nz_{n+1},q,q^{s+2};q,p)_t}\\
&\qquad\cdot\frac{(q^{-s}/a_1z_{n+1},\dots,q^{-s}/a_{m+n+1}z_{n+1},q^{-1};q,p)_u}{(z_1q^{-s}/z_{n+1},\dots,z_nq^{-s}/z_{n+1},q^{1-s}/w_1z_{n+1},\dots,q^{1-s}/w_nz_{n+1},q^{-s},q;q,p)_u}\\
&=\sum_{\substack{0\leq y_1,\dots,y_m,t\\y_1+\cdots+y_m\leq N-t}}\sum_{\substack{0\leq s,u\\s+u=N-t-y_1-\cdots-y_m}}\frac{(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_s}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}/z_1,\dots,z_{n+1}/z_{n},q;q,p)_s}\\
&\qquad\cdot\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m},q^{t+1}/z_{n+1},q^{u-s}/z_{n+1};p)}{\Delta(w_1,\dots,w_m,q/z_{n+1},q^{-s}/z_{n+1};p)}\\
&\qquad\cdot\prod_{k=1}^m\frac{(w_k/a_1,\dots,w_k/a_{m+n+1},w_kz_{n+1}q^{s-1};q,p)_{y_k}}{(w_kz_1,\dots,w_kz_{n+1},w_kq/w_1,\dots,w_kq/w_m,w_kz_{n+1}q^{s+1};q,p)_{y_k}}\\
&\qquad\cdot\frac{(q/a_1z_{n+1},\dots,q/a_{m+n+1}z_{n+1},q^{s};q,p)_t}{(z_1q/z_{n+1},\dots,z_nq/z_{n+1},q^2/w_1z_{n+1},\dots,q^2/w_nz_{n+1},q,q^{s+2};q,p)_t}\\
&\qquad\cdot\frac{(q^{-s}/a_1z_{n+1},\dots,q^{-s}/a_{m+n+1}z_{n+1},q^{-1};q,p)_u}{(z_1q^{-s}/z_{n+1},\dots,z_nq^{-s}/z_{n+1},q^{1-s}/w_1z_{n+1},\dots,q^{1-s}/w_nz_{n+1},q^{-s},q;q,p)_u}
\end{align}
となる. ここで, $(q^{-1};q,p)_u$が掛かっていることによって, $u\in\{0,1\}$の場合以外は$0$になる. $u\in\{0,1\}$であるとき,
\begin{align}
(A;q,p)_s(q^{-s}/A;q,p)_u&=(-q^{-s}/A)^u(A;q,p)_{s+u}\\
\end{align}
が成り立つことから, 和の$s,u$に依存する因子だけを集めると
\begin{align}
&\frac{(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_s}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}/z_1,\dots,z_{n+1}/z_{n},q;q,p)_s}\\
&\qquad\cdot\left(\prod_{j=1}^m\frac{w_jq^{y_j}\theta(q^{u-s-y_j}/w_jz_{n+1};p)}{w_j\theta(q^{-s}/w_jz_{n+1};p)}\right)\frac{(q^{t+1}/z_{n+1})\theta(q^{u-s-t-1};p)}{(q/z_{n+1})\theta(q^{-s-1};p)}\\
&\qquad\cdot\left(\prod_{k=1}^m\frac{(w_kz_{n+1}q^{s-1};q,p)_{y_k}}{(w_kz_{n+1}q^{s+1};q,p)_{y_k}}\right)\frac{(q^s;q,p)_t}{(q^{s+2};q,p)_t}\\
&\qquad\cdot\frac{(q^{-s}/a_1z_{n+1},\dots,q^{-s}/a_{m+n+1}z_{n+1},q^{-1};q,p)_u}{(z_1q^{-s}/z_{n+1},\dots,z_nq^{-s}/z_{n+1},q^{1-s}/w_1z_{n+1},\dots,q^{1-s}/w_nz_{n+1},q^{-s},q;q,p)_u}\\
&=\frac{(-q^{-s})^u(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_{s+u}}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}/z_1,\dots,z_{n+1}/z_{n};q,p)_{s+u}(q;q,p)_s}\\
&\qquad\cdot\left(\prod_{j=1}^mq^u\frac{\theta(w_jz_{n+1}q^{s-1},w_jz_{n+1}q^{s-u+y_j},q^{-s}/w_jz_{n+1};p)}{\theta(w_jz_{n+1}q^{s+y_j-1},w_jz_{n+1}q^{s+y_j},q^{u-s}/w_jz_{n+1};p)}\right)q^u\frac{\theta(q^{s+t-u+1};p)}{\theta(q^{s+1};p)}\frac{(q^s;q,p)_t}{(q^{s+2};q,p)_t}\frac{(q^{-1};q,p)_u}{(q^{-s},q;q,p)_u}\\
&=\frac{(-1)^u(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_{s+u}}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}/z_1,\dots,z_{n+1}/z_{n};q,p)_{s+u}}\\
&\qquad\cdot\left(\prod_{j=1}^m\frac{\theta(w_jz_{n+1}q^{s+u-1};p)}{\theta(w_jz_{n+1}q^{s+u+y_j-1};p)}\right)\frac{\theta(q^{s+t-u+1};p)(q^s;q,p)_t}{(q;q,p)_{s+t+1}(q^{s};q,p)_u}\\
&=\frac{(-1)^u(a_1z_{n+1},\dots,a_{m+n+1}z_{n+1};q,p)_{s+u}}{(w_1z_{n+1},\dots,w_mz_{n+1},z_{n+1}/z_1,\dots,z_{n+1}/z_{n};q,p)_{s+u}}\\
&\qquad\cdot\left(\prod_{j=1}^m\frac{\theta(w_jz_{n+1}q^{s+u-1};p)}{\theta(w_jz_{n+1}q^{s+u+y_j-1};p)}\right)\frac{(q^{s+u};q,p)_t}{(q;q,p)_{s+t+u}}
\end{align}
となる. これは$(-1)^u$の因子以外は$s+u$にのみ依存しているから, $s+u>0$のとき, $s+u$を固定して$u\in\{0,1\}$について和をとると, $0$になることが分かる. よって, $s+u=N-t-y_1-\cdots-y_m=0$の場合だけが残ることが分かる. このとき, さらに上の$(q^{s+u};q,p)_t$は$t>0$のとき$0$になるから, $t=0$の場合だけが残ることが分かる. よって,
\begin{align}
&\sum_{\substack{0\leq y_1,\dots,y_{n+1}\\y_1+\cdots+y_{n+1}=N}}\frac{\Delta(z_1q^{y_1},\dots,z_{n+1}q^{y_{n+1}};p)}{\Delta(z_1,\dots,z_{n+1};p)}\prod_{k=1}^{n+1}\frac{(a_1z_k,\dots,a_{m+n+1}z_k;q,p)_{y_k}}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_{n+1};q,p)_{y_k}}\\
&=\sum_{\substack{0\leq y_1,\dots,y_m\\y_1+\cdots+y_m=N}}\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m},q/z_{n+1},1/z_{n+1};p)}{\Delta(w_1,\dots,w_m,q/z_{n+1},1/z_{n+1};p)}\\
&\qquad\cdot\prod_{k=1}^m\frac{(w_k/a_1,\dots,w_k/a_{m+n+1},w_kz_{n+1}/q;q,p)_{y_k}}{(w_kz_1,\dots,w_kz_{n+1},w_kq/w_1,\dots,w_kq/w_m,w_kz_{n+1}q;q,p)_{y_k}}\\
&=\sum_{\substack{0\leq y_1,\dots,y_m\\y_1+\cdots+y_m=N}}\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m},q/z_{n+1},1/z_{n+1};p)}{\Delta(w_1,\dots,w_m,q/z_{n+1},1/z_{n+1};p)}\\
&\qquad\cdot\prod_{k=1}^m\frac{\theta(w_kz_{n+1}q^{y_k-1},w_kz_{n+1}q^{y_k};p)(w_k/a_1,\dots,w_k/a_{m+n+1},w_kz_{n+1}/q;q,p)_{y_k}}{\theta(w_kz_{n+1}/q,w_kz_{n+1};p)(w_kz_1,\dots,w_kz_{n+1},w_kq/w_1,\dots,w_kq/w_m,w_kz_{n+1}q;q,p)_{y_k}}\\
&=\sum_{\substack{0\leq y_1,\dots,y_m\\y_1+\cdots+y_m=N}}\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m},q/z_{n+1},1/z_{n+1};p)}{\Delta(w_1,\dots,w_m,q/z_{n+1},1/z_{n+1};p)}\\
&\qquad\cdot\prod_{k=1}^m\frac{(w_k/a_1,\dots,w_k/a_{m+n+1};q,p)_{y_k}}{(w_kz_1,\dots,w_kz_{n+1},w_kq/w_1,\dots,w_kq/w_m;q,p)_{y_k}}
\end{align}
となって示すべき等式が得られる.
この定理1は, 前の記事 で示したFrenkel-TuraevによるBaileyの変換公式の楕円類似を一般化している. 実際, $n=m=2$としてパラメータを置き換えることによってそれを得ることができる.
定理1において, 特に$p=0$とすると以下の変換公式を得る. それはKajiharaによって2003年に示された結果である.
$m,n\geq 1, w_1\cdots w_m=a_1\cdots a_{m+n}z_1\cdots z_n$であるとき,
\begin{align}
&\sum_{\substack{0\leq y_1,\dots,y_n\\y_1+\cdots+y_n=N}}\frac{\Delta(z_1q^{y_1},\dots,z_nq^{y_n})}{\Delta(z_1,\dots,z_n)}\prod_{k=1}^n\frac{(a_1z_k,\dots,a_{m+n}z_k;q)_{y_k}}{(w_1z_k,\dots,w_mz_k,z_kq/z_1,\dots,z_kq/z_n;q)_{y_k}}\\
&=\sum_{\substack{0\leq y_1,\dots,y_m\\y_1+\cdots+y_m=N}}\frac{\Delta(w_1q^{y_1},\dots,w_mq^{y_m})}{\Delta(w_1,\dots,w_m)}\prod_{k=1}^m\frac{(w_k/a_1,\dots,w_k/a_{m+n};q)_{y_k}}{(w_kz_1,\dots,w_kz_n,w_kq/w_1,\dots,w_kq/w_m;q)_{y_k}}
\end{align}
が成り立つ.
