Baileyによる10φ9の4項変換公式

前回の記事 で示した等式
\begin{align} &\int_a^b\frac{(t/\sqrt a,-t/\sqrt a,tq/a,tq/b,tq/c,tq/d,tq/e,tq/f;q)_{\infty}}{(tq/\sqrt a,-tq/\sqrt a,t,bt/a,ct/a,dt/a,et/a,ft/a;q)_{\infty}}\,d_qt\\ &=b\frac{(q,a/b,bq/a,aq/cd,aq/ce,aq/cf,aq/de,aq/df,aq/ef;q)_{\infty}}{(b,c,d,e,f,bc/a,bd/a,be/a,bf/a;q)_{\infty}}\qquad(a^2q=bcdef) \end{align}
において, $w=a^2q/cde, a\mapsto w, b\mapsto bq^n,c\mapsto wc/a,d\mapsto wd/a,e\mapsto we/a,f\mapsto a/bq^n$として,
\begin{align} &\int_w^{bq^n}\frac{(t/\sqrt w,-t/\sqrt w,tq/w,tq^{1-n}/b,atq/wc,atq/wd,atq/we,btq^{n+1}/a;q)_{\infty}}{(tq/\sqrt w,-tq/\sqrt w,t,btq^{n}/w,ct/a,dt/a,et/a,atq^{-n}/wb)_{\infty}}\,d_qt\\ &=bq^n\frac{(q,wq^{-n}/b,bq^{n+1}/w,a^2q/wcd,a^2q/wce,a^2q/wde,bq^{n+1}/c,bq^{n+1}/d,bq^{n+1}/e;q)_{\infty}}{(bq^n,wc/a,wd/a,we/a,a/bq^n,bcq^n/a,bdq^n/a,beq^n/a,a/w;q)_{\infty}}\\ &=b\frac{(q,w/b,bq/w,c,d,e,bq/c,bq/d,bq/e;q)_{\infty}}{(b,wc/a,wd/a,we/a,a/b,bc/a,bd/a,be/a,a/w;q)_{\infty}}\frac{(b,bc/a,bd/a,be/a;q)_n(a/b;q)_{-n}}{(bq/w,bq/c,bq/d,bq/e;q)_n(w/b;q)_{-n}}q^n\\ &=b\frac{(q,w/b,bq/w,c,d,e,bq/c,bq/d,bq/e;q)_{\infty}}{(b,wc/a,wd/a,we/a,a/b,bc/a,bd/a,be/a,a/w;q)_{\infty}}\frac{(b,bc/a,bd/a,be/a;q)_n}{(bq/a,bq/c,bq/d,bq/e;q)_n}\left(\frac{wq}{a}\right)^n\\ \end{align}
を得る. $v=a^3q^2/bcdefgh$として, 両辺に
\begin{align} \frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n \end{align}
を掛けて足し合わせると, 右辺は
\begin{align} b\frac{(q,w/b,bq/w,c,d,e,bq/c,bq/d,bq/e;q)_{\infty}}{(b,wc/a,wd/a,we/a,a/b,bc/a,bd/a,be/a,a/w;q)_{\infty}}W(b^2/a,b,bc/a,bd/a,be/a,bf/a,bg/a,bh/a;vq) \end{align}
となる. 左辺は,
\begin{align} &\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n\\ &\qquad \cdot \,\int_w^{bq^n}\frac{(t/\sqrt w,-t/\sqrt w,tq/w,tq^{1-n}/b,atq/wc,atq/wd,atq/we,btq^{n+1}/a;q)_{\infty}}{(tq/\sqrt w,-tq/\sqrt w,t,btq^{n}/w,ct/a,dt/a,et/a,atq^{-n}/wb)_{\infty}}\,d_qt\\ &=\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n\\ &\qquad\cdot \sum_{0\leq m}bq^{n+m}\frac{(bq^{n+m}/\sqrt w,-bq^{n+m}/\sqrt w,bq^{n+m+1}/w,q^{1+m},abq^{n+m+1}/wc,abq^{n+m+1}/wd,abq^{n+m+1}/we,b^2q^{2n+m+1}/a;q)_{\infty}}{(bq^{n+m+1}/\sqrt w,-bq^{n+m+1}/\sqrt w,bq^{n+m},b^2q^{2n+m}/w,bcq^{n+m}/a,bdq^{n+m}/a,beq^{n+m}/a,aq^{m}/w)_{\infty}}\\ &\qquad -\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n\\ &\qquad \cdot \sum_{0\leq m}wq^m\frac{(\sqrt wq^m,-\sqrt wq^m,q^{m+1},wq^{1+m-n}/b,aq^{m+1}/c,aq^{m+1}/d,aq^{m+1}/e,wbq^{n+m+1}/a;q)_{\infty}}{(\sqrt wq^{m+1},-\sqrt wq^{m+1},wq^{m},bq^{n+m},wcq^{m}/a,wdq^{m}/a,weq^{m}/a,aq^{m-n}/b)_{\infty}} \end{align}
1つ目の項は,
\begin{align} &\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n\\ &\qquad \cdot \sum_{0\leq m}bq^{n+m}\frac{(bq^{n+m}/\sqrt w,-bq^{n+m}/\sqrt w,bq^{n+m+1}/w,q^{1+m},abq^{n+m+1}/wc,abq^{n+m+1}/wd,abq^{n+m+1}/we,b^2q^{2n+m}/a;q)_{\infty}}{(bq^{n+m+1}/\sqrt w,-bq^{n+m+1}/\sqrt w,bq^{n+m},b^2q^{2n+m+1}/w,bcq^{n+m}/a,bdq^{n+m}/a,beq^{n+m}/a,aq^{m}/w)_{\infty}}\\ &=b\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n\\ &\qquad\cdot \sum_{n\leq m}q^{m}\frac{(bq^{m}/\sqrt w,-bq^{m}/\sqrt w,bq^{m+1}/w,q^{1+m-n},abq^{m+1}/wc,abq^{m+1}/wd,abq^{m+1}/we,b^2q^{n+m+1}/a;q)_{\infty}}{(bq^{m+1}/\sqrt w,-bq^{m+1}/\sqrt w,bq^{m},b^2q^{n+m}/w,bcq^{m}/a,bdq^{m}/a,beq^{m}/a,aq^{m-n}/w)_{\infty}}\\ &=b\frac{(b/\sqrt w,-b/\sqrt w,bq/w,q,abq/wc,abq/wd,abq/we,b^2q/a;q)_{\infty}}{(bq/\sqrt w,-bq/\sqrt w,b,b^2/w,bc/a,bd/a,be/a,a/w;q)_{\infty}}\\ &\qquad\cdot \sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n\\ &\qquad\qquad\cdot \sum_{n\leq m}q^{m}\frac{(bq/\sqrt w,-bq/\sqrt w,b,bc/a,bd/a,be/a;q)_m(b^2/w;q)_{n+m}(a/w;q)_{m-n}}{(b/\sqrt w,-b/\sqrt w,abq/wc,abq/wd,abq/we;q)_m(b^2q/a;q)_{n+m}(q;q)_{m-n}}\\ &=b\frac{(bq/w,q,abq/wc,abq/wd,abq/we,b^2q/a;q)_{\infty}}{(b,b^2q/w,bc/a,bd/a,be/a,a/w;q)_{\infty}}\sum_{0\leq m}q^{m}\frac{(bq/\sqrt w,-bq/\sqrt w,b,bc/a,bd/a,be/a;q)_m}{(b/\sqrt w,-b/\sqrt w,abq/wc,abq/wd,abq/we;q)_m}\\ &\qquad\cdot\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\frac{(b^2/w;q)_{n+m}(a/w;q)_{m-n}}{(b^2q/a;q)_{n+m}(q;q)_{m-n}}\left(\frac{av}{w}\right)^n\\ &=b\frac{(bq/w,q,abq/wc,abq/wd,abq/we,b^2q/a;q)_{\infty}}{(b,b^2q/w,bc/a,bd/a,be/a,a/w;q)_{\infty}}\sum_{0\leq m}q^{m}\frac{(bq/\sqrt w,-bq/\sqrt w,b,bc/a,bd/a,be/a,b^2/w,a/w;q)_m}{(b/\sqrt w,-b/\sqrt w,abq/wc,abq/wd,abq/we,b^2q/a,q;q)_m}\\ &\qquad\cdot \sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a,b^2q^m/w,q^{-m};q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,b^2q^{m+1}/a,wq^{1-m}/a,q;q)_n}\left(vq\right)^n\\ \end{align}
ここで, $v=1$とすると, Jacksonの和公式 より,
\begin{align} \sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a,b^2q^m/w,q^{-m};q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,b^2q^{m+1}/a,wq^{1-m}/a,q;q)_n}\left(vq\right)^n&=\frac{(b^2q/a,aq/fg,aq/fh,aq/gh;q)_m}{(bq/f,bq/g,bq/h,a/w;q)_{m}}\\ &=\frac{(b^2q/a,bf/w,bg/w,bh/w;q)_m}{(bq/f,bq/g,bq/h,a/w;q)_{m}} \end{align}
だから,
\begin{align} &\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{av}{w}\right)^n\\ &\qquad\cdot \sum_{0\leq m}bq^{n+m}\frac{(bq^{n+m}/\sqrt w,-bq^{n+m}/\sqrt w,bq^{n+m+1}/w,q^{1+m},abq^{n+m+1}/wc,abq^{n+m+1}/wd,abq^{n+m+1}/we,b^2q^{2n+m}/a;q)_{\infty}}{(bq^{n+m+1}/\sqrt w,-bq^{n+m+1}/\sqrt w,bq^{n+m},b^2q^{2n+m+1}/w,bcq^{n+m}/a,bdq^{n+m}/a,beq^{n+m}/a,aq^{m}/w)_{\infty}}\\ &=b\frac{(bq/w,q,abq/wc,abq/wd,abq/we,b^2q/a;q)_{\infty}}{(b,b^2q/w,bc/a,bd/a,be/a,a/w;q)_{\infty}}\\ &\qquad\cdot\sum_{0\leq m}q^{m}\frac{(bq/\sqrt w,-bq/\sqrt w,b,bc/a,bd/a,be/a,b^2/w,a/w;q)_m}{(b/\sqrt w,-b/\sqrt w,abq/wc,abq/wd,abq/we,b^2q/a,q;q)_m}\frac{(b^2q/a,aq/fg,aq/fh,aq/gh;q)_m}{(bq/f,bq/g,bq/h,a/w;q)_{m}}\\ &=b\frac{(bq/w,q,abq/wc,abq/wd,abq/we,b^2q/a;q)_{\infty}}{(b,b^2q/w,bc/a,bd/a,be/a,a/w;q)_{\infty}}W(b^2/w,b,bc/a,bd/a,be/a,bf/w,bg/w,bh/w;q) \end{align}
である. 次に2つ目の項の符号を除いた部分は, $v=1$として,
\begin{align} &\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{a}{w}\right)^n\\ &\qquad\cdot\sum_{0\leq m}wq^m\frac{(\sqrt wq^m,-\sqrt wq^m,q^{m+1},wq^{1+m-n}/b,aq^{m}/c,aq^{m}/d,aq^{m}/e,wbq^{n+m+1}/a;q)_{\infty}}{(\sqrt wq^{m+1},-\sqrt wq^{m+1},wq^{m+1},bq^{n+m},wcq^{m+1}/a,wdq^{m+1}/a,weq^{m+1}/a,aq^{m-n}/b)_{\infty}}\\ &=w\frac{(\sqrt w,-\sqrt w,q,wq/b,aq/c,aq/d,aq/e,wbq/a;q)_{\infty}}{(\sqrt wq,-\sqrt w,w,b,wc/a,wd/a,we/a,a/b;q)_{\infty}}\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{a}{w}\right)^n\sum_{0\leq m}q^m\frac{(\sqrt wq,-\sqrt wq,w,wc/a,wd/a,we/a;q)_m(b;q)_{n+m}(a/b;q)_{m-n}}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,q;q)_m(wbq/a;q)_{n+m}(wq/b;q)_{m-n}}\\ &=w\frac{(q,wq/b,aq/c,aq/d,aq/e,wbq/a;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,a/b;q)_{\infty}}\sum_{0\leq m}q^m\frac{(\sqrt wq,-\sqrt wq,w,wc/a,wd/a,we/a;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,q;q)_m}\\ &\qquad\cdot\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\frac{(b;q)_{n+m}(a/b;q)_{m-n}}{(wbq/a;q)_{n+m}(wq/b;q)_{m-n}}\left(\frac{a}{w}\right)^n\\ &=w\frac{(q,wq/b,aq/c,aq/d,aq/e,wbq/a;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,a/b;q)_{\infty}}\sum_{0\leq m}q^m\frac{(\sqrt wq,-\sqrt wq,w,wc/a,wd/a,we/a,b,a/b;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,wbq/a,wq/b,q;q)_m}\\ &\qquad\cdot\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a,bq^m,bq^{-m}/w;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,wbq^{m+1}/a,bq^{1-m}/a,q;q)_n}q^n\\ \end{align}
ここで, Non-terminating Jacksonの和公式
\begin{align} &W\left(a;b,c,d,e,f;q\right)-\frac ba\frac{(aq,bq/a,bq/c,bq/d,bq/e,bq/f,c,d,e,f;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\,W\left(b^2/a;b,bc/a,bd/a,be/a,bf/a;q\right)\\ &=\frac{(aq,aq/cd,aq/ce,aq/cf,aq/de,aq/df,aq/ef,b/a;q)_{\infty}}{(aq/c,aq/d,aq/e,aq/f,bc/a,bd/a,be/a,bf/a;q)_{\infty}}\qquad(a^2q=bcdef) \end{align}
より, 上の$b$に$bq^m$を選んで,
\begin{align} &\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a,bq^m,bq^{-m}/w;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,wbq^{m+1}/a,bq^{1-m}/a,q;q)_n}q^n\\ &=\frac{(b^2q/a,aq/fg,aq/fh,aq/gh,wq^{m+1}/f,wq^{m+1}/g,wq^{m+1}/h,aq^m/b;q)_{\infty}}{(bq/f,bq/g,bq/h,wbq^{m+1}/a,fq^m,gq^m,hq^m,a/w;q)_{\infty}}\\ &\qquad+\frac{aq^m}b\frac{(b^2q/a,aq^{m+1}/b,aq^{m+1}/f,aq^{m+1}/g,aq^{m+1}/h,wq^{2m+1},bf/a,bg/abh/a,bq^{-m}/w;q)_{\infty}}{(bq^{1-m}/a,bq/f,bq/g,bq/h,wbq^{m+1}/a,fq^m,gq^m,hq^m,a/w,aq^{2m+1};q)_{\infty}}\\ &\qquad\qquad W(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q)\\ &=\frac{(b^2q/a,aq/fg,aq/fh,aq/gh,wq/f,wq/g,wq/h,a/b;q)_{\infty}}{(bq/f,bq/g,bq/h,wbq/a,f,g,h,a/w;q)_{\infty}}\frac{(wbq/a,f,g,h;q)_m}{(wq/f,wq/g,wq/h,a/b;q)_m}\\ &\qquad+\frac{a}b\frac{(b^2q/a,aq/b,aq/f,aq/g,aq/h,wq,bf/a,bg/a,bh/a,b/w;q)_{\infty}}{(bq/a,bq/f,bq/g,bq/h,wbq/a,f,g,h,a/w,aq;q)_{\infty}}\frac{(wbq/a,f,g,h)_m(aq;q)_{2m}(bq/a;q)_{-m}}{(aq/b,aq/f,aq/g,aq/h;q)_m(wq;q)_{2m}(b/w;q)_{-m}}q^m\\ &\qquad\qquad W(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q)\\ &=\frac{(b^2q/a,aq/fg,aq/fh,aq/gh,wq/f,wq/g,wq/h,a/b;q)_{\infty}}{(bq/f,bq/g,bq/h,wbq/a,f,g,h,a/w;q)_{\infty}}\frac{(wbq/a,f,g,h;q)_m}{(wq/f,wq/g,wq/h,a/b;q)_m}\\ &\qquad+\frac{a}b\frac{(b^2q/a,aq/b,aq/f,aq/g,aq/h,wq,bf/a,bg/a,bh/a,b/w;q)_{\infty}}{(bq/a,bq/f,bq/g,bq/h,wbq/a,f,g,h,a/w,aq;q)_{\infty}}\\ &\qquad\cdot\frac{(wbq/a,f,g,h,wq/b;q)_m(aq;q)_{2m}}{(aq/b,aq/f,aq/g,aq/h,a/b;q)_m(wq;q)_{2m}}\left(\frac{a}{w}\right)^mW(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q) \end{align}
これより,
\begin{align} &\sum_{0\leq n}\frac{(1-b^2q^{2n}/a)(b^2/a,bf/a,bg/a,bh/a;q)_n}{(1-b^2/a)(bq/f,bq/g,bq/h,q;q)_n}\left(\frac{a}{w}\right)^n\\ &\qquad\cdot\sum_{0\leq m}wq^m\frac{(\sqrt wq^m,-\sqrt wq^m,q^{m+1},wq^{1+m-n}/b,aq^{m}/c,aq^{m}/d,aq^{m}/e,wbq^{n+m+1}/a;q)_{\infty}}{(\sqrt wq^{m+1},-\sqrt wq^{m+1},wq^{m+1},bq^{n+m},wcq^{m+1}/a,wdq^{m+1}/a,weq^{m+1}/a,aq^{m-n}/b)_{\infty}}\\ &=w\frac{(q,wq/b,aq/c,aq/d,aq/e,wbq/a;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,a/b;q)_{\infty}}\sum_{0\leq m}q^m\frac{(\sqrt wq,-\sqrt wq,w,wc/a,wd/a,we/a,b,a/b;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,wbq/a,wq/b,q;q)_m}\\ &\qquad\cdot\frac{(b^2q/a,aq/fg,aq/fh,aq/gh,wq/f,wq/g,wq/h,a/b;q)_{\infty}}{(bq/f,bq/g,bq/h,wbq/a,f,g,h,a/w;q)_{\infty}}\frac{(wbq/a,f,g,h;q)_m}{(wq/f,wq/g,wq/h,a/b;q)_m}\\ &\qquad+w\frac{(q,wq/b,aq/c,aq/d,aq/e,wbq/a;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,a/b;q)_{\infty}}\sum_{0\leq m}\frac{(\sqrt wq,-\sqrt wq,w,wc/a,wd/a,we/a,b,a/b;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,wbq/a,wq/b,q;q)_m}\\ &\qquad\cdot\frac{a}b\frac{(b^2q/a,aq/b,aq/f,aq/g,aq/h,wq,bf/a,bg/a,bh/a,b/w;q)_{\infty}}{(bq/a,bq/f,bq/g,bq/h,wbq/a,f,g,h,a/w,aq;q)_{\infty}}\\ &\qquad\cdot \frac{(wbq/a,f,g,h,wq/b;q)_m(aq;q)_{2m}}{(aq/b,aq/f,aq/g,aq/h,a/b;q)_m(wq;q)_{2m}}\left(\frac{aq}{w}\right)^mW(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q) \end{align}
この1つ目の項は,
\begin{align} &w\frac{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/fg,aq/fh,aq/gh,wq/f,wq/g,wq/h;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,bq/f,bq/g,bq/h,f,g,h,a/w;q)_{\infty}}\\ &\qquad\cdot\sum_{0\leq m}q^m\frac{(\sqrt wq,-\sqrt wq,w,wc/a,wd/a,we/a,b,f,g,h;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,wq/b,wq/f,wq/g,wq/h,q;q)_m}\\ &=w\frac{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/fg,aq/fh,aq/gh,wq/f,wq/g,wq/h;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,bq/f,bq/g,bq/h,f,g,h,a/w;q)_{\infty}}\\ &\qquad\cdot W(w;b,wc/a,wd/a,we/a,f,g,h;q) \end{align}
であり, 2つ目の項は,
\begin{align} &w\frac{(q,wq/b,aq/c,aq/d,aq/e,wbq/a;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,a/b;q)_{\infty}}\sum_{0\leq m}\frac{(\sqrt wq,-\sqrt wq,w,wc/a,wd/a,we/a,b,a/b;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,wbq/a,wq/b,q;q)_m}\\ &\qquad\cdot\frac{a}b\frac{(b^2q/a,aq/b,aq/f,aq/g,aq/h,wq,bf/a,bg/a,bh/a,b/w;q)_{\infty}}{(bq/a,bq/f,bq/g,bq/h,wbq/a,f,g,h,a/w,aq;q)_{\infty}}\\ &\qquad\qquad\cdot \frac{(wbq/a,f,g,h,wq/b;q)_m(aq;q)_{2m}}{(aq/b,aq/f,aq/g,aq/h,a/b;q)_m(wq;q)_{2m}}\left(\frac{aq}{w}\right)^mW(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q)\\ &=\frac{aw}b\frac{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/b,aq/f,aq/g,aq/h,bf/a,bg/a,bh/a,b/w;q)_{\infty}}{(b,wc/a,wd/a,we/a,a/b,bq/a,bq/f,bq/g,bq/h,f,g,h,a/w,aq;q)_{\infty}}\\ &\qquad\cdot\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a,b,f,g,h;q)_m}{(\sqrt w,-\sqrt w,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,q;q)_m}\frac{(aq;q)_{2m}}{(wq;q)_{2m}}\left(\frac{aq}{w}\right)^m\\ &\qquad\qquad \cdot W(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q) \end{align}
ここで, 和の部分は
\begin{align} &\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a,b,f,g,h;q)_m}{(\sqrt w,-\sqrt w,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,q;q)_m}\frac{(aq;q)_{2m}}{(wq;q)_{2m}}\left(\frac{aq}{w}\right)^m\\ &\qquad \cdot W(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q)\\ &=\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a,b,f,g,h;q)_m}{(\sqrt w,-\sqrt w,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,q;q)_m}\frac{(aq;q)_{2m}}{(wq;q)_{2m}}\left(\frac{aq}{w}\right)^m\\ &\qquad\sum_{0\leq n}\frac{(aq^{2m},\sqrt aq^{m+1},-\sqrt aq^{m+1},bq^m,fq^m,gq^m,hq^m,a/w;q)_n}{(\sqrt aq^m,-\sqrt aq^m,aq^{m+1}/b,aq^{m+1}/f,aq^{m+1}/g,aq^{m+1}/h,wq^{2m+1};q)_n}q^n\\ &=\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,q;q)_m}\left(\frac{aq}{w}\right)^m\\ &\qquad\sum_{0\leq n}\frac{(a;q)_{n+2m}(1-aq^{2m+2n})(b,f,g,h;q)_{n+m}(a/w;q)_n}{(1-a)(aq/b,aq/f,aq/g,aq/h;q)_{n+m}(q;q)_n(wq;q)_{n+2m}}q^n\\ &=\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,q;q)_m}\left(\frac{a}{w}\right)^m\\ &\qquad\cdot\sum_{m\leq n}\frac{(a;q)_{n+m}(1-aq^{2n})(b,f,g,h;q)_{n}(a/w;q)_{n-m}}{(1-a)(aq/b,aq/f,aq/g,aq/h;q)_{n}(q;q)_{n-m}(wq;q)_{n+m}}q^n\\ &=\sum_{0\leq n}\frac{(1-aq^{2n})(b,f,g,h;q)_{n}}{(1-a)(aq/b,aq/f,aq/g,aq/h;q)_{n}}q^n\\ &\qquad\cdot\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a;q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,q;q)_m}\frac{(a;q)_{n+m}(a/w;q)_{n-m}}{(wq;q)_{n+m}(q;q)_{n-m}}\left(\frac{a}{w}\right)^m\\ &=\sum_{0\leq n}\frac{(1-aq^{2n})(a,b,f,g,h,a/w;q)_{n}}{(1-a)(aq/b,aq/f,aq/g,aq/h,wq,q;q)_{n}}q^n\\ &\qquad\cdot\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a,aq^n,q^{-n};q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,wq^{n+1},wq^{1-n}/a,q;q)_m}q^m \end{align}
ここで, Jacksonの和公式より,
\begin{align} \sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a,aq^n,q^{-n};q)_m}{(\sqrt w,-\sqrt w,aq/c,aq/d,aq/e,wq^{n+1},wq^{1-n}/a,q;q)_m}q^m&=\frac{(wq,c,d,e;q)_n}{(aq/c,aq/d,aq/e,a/w;q)_n} \end{align}
であるから,
\begin{align} &\sum_{0\leq m}\frac{(w,\sqrt wq,-\sqrt wq,wc/a,wd/a,we/a,b,f,g,h;q)_m}{(\sqrt w,-\sqrt w,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,q;q)_m}\frac{(aq;q)_{2m}}{(wq;q)_{2m}}\left(\frac{aq}{w}\right)^m\\ &\qquad \cdot W(aq^{2m};bq^m,fq^m,gq^m,hq^m,a/w;q)\\ &=\sum_{0\leq n}\frac{(1-aq^{2n})(a,b,f,g,h,a/w;q)_{n}}{(1-a)(aq/b,aq/f,aq/g,aq/h,wq,q;q)_{n}}q^n\frac{(wq,c,d,e;q)_n}{(aq/c,aq/d,aq/e,a/w;q)_n}\\ &=\sum_{0\leq n}\frac{(1-aq^{2n})(a,b,c,d,e,f,g,h;q)_{n}}{(1-a)(aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,q;q)_{n}}q^n\\ &=W(a;b,c,d,e,f,g,h;q) \end{align}
である. よって, これまでの計算を全てまとめると, $a^3q^2=bcdefgh$のとき,
\begin{align} &b\frac{(q,w/b,bq/w,c,d,e,bq/c,bq/d,bq/e;q)_{\infty}}{(b,wc/a,wd/a,we/a,a/b,bc/a,bd/a,be/a,a/w;q)_{\infty}}W(b^2/a,b,bc/a,bd/a,be/a,bf/a,bg/a,bh/a;q)\\ &=b\frac{(bq/w,q,abq/wc,abq/wd,abq/we,b^2q/a;q)_{\infty}}{(b,b^2q/w,bc/a,bd/a,be/a,a/w;q)_{\infty}}W(b^2/w,b,bc/a,bd/a,be/a,bf/w,bg/w,bh/w;q)\\ &-w\frac{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/fg,aq/fh,aq/gh,wq/f,wq/g,wq/h;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,bq/f,bq/g,bq/h,f,g,h,a/w;q)_{\infty}}\\ &\qquad\cdot W(w;b,wc/a,wd/a,we/a,f,g,h;q)\\ &-\frac{aw}b\frac{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/b,aq/f,aq/g,aq/h,bf/a,bg/a,bh/a,b/w;q)_{\infty}}{(b,wc/a,wd/a,we/a,a/b,bq/a,bq/f,bq/g,bq/h,f,g,h,a/w,aq;q)_{\infty}}W(a;b,c,d,e,f,g,h;q) \end{align}
だから,
\begin{align} &W(a;b,c,d,e,f,g,h;q)\\ &=-\frac{b}{aw}\frac{(b,wc/a,wd/a,we/a,a/b,bq/a,bq/f,bq/g,bq/h,f,g,h,a/w,aq;q)_{\infty}}{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/b,aq/f,aq/g,aq/h,bf/a,bg/a,bh/a,b/w;q)_{\infty}}\\ &\qquad\cdot\,b\frac{(q,w/b,bq/w,c,d,e,bq/c,bq/d,bq/e;q)_{\infty}}{(b,wc/a,wd/a,we/a,a/b,bc/a,bd/a,be/a,a/w;q)_{\infty}}W(b^2/a;b,bc/a,bd/a,be/a,bf/a,bg/a,bh/a;q)\\ &+\frac{b}{aw}\frac{(b,wc/a,wd/a,we/a,a/b,bq/a,bq/f,bq/g,bq/h,f,g,h,a/w,aq;q)_{\infty}}{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/b,aq/f,aq/g,aq/h,bf/a,bg/a,bh/a,b/w;q)_{\infty}}\\ &\qquad\cdot b\frac{(bq/w,q,abq/wc,abq/wd,abq/we,b^2q/a;q)_{\infty}}{(b,b^2q/w,bc/a,bd/a,be/a,a/w;q)_{\infty}}W(b^2/w;b,bc/a,bd/a,be/a,bf/w,bg/w,bh/w;q)\\ &-\frac{b}{aw}\frac{(b,wc/a,wd/a,we/a,a/b,bq/a,bq/f,bq/g,bq/h,f,g,h,a/w,aq;q)_{\infty}}{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/b,aq/f,aq/g,aq/h,bf/a,bg/a,bh/a,b/w;q)_{\infty}}\\ &\qquad\cdot w\frac{(q,wq/b,aq/c,aq/d,aq/e,b^2q/a,aq/fg,aq/fh,aq/gh,wq/f,wq/g,wq/h;q)_{\infty}}{(wq,b,wc/a,wd/a,we/a,bq/f,bq/g,bq/h,f,g,h,a/w;q)_{\infty}}\\ &\qquad\cdot W(w;b,wc/a,wd/a,we/a,f,g,h;q)\\ &=-\frac{(b/a,bq/c,bq/d,bq/e,bq/f,bq/g,bq/h,c,d,e,f,g,h,aq;q)_{\infty}}{(a/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,b^2q/a,bc/a,bd/a,be/a,bf/a,bg/a,bh/a;q)_{\infty}}\\ &\qquad\cdot\,W(b^2/a;b,bc/a,bd/a,be/a,bf/a,bg/a,bh/a;q)\\ &+\frac{(wc/a,wd/a,we/a,b/a,bq/f,bq/g,bq/h,f,g,h,aq,abq/wc,abq/wd,abq/we;q)_{\infty}}{(w/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq/h,bc/a,bd/a,be/a,bf/a,bg/a,bh/a,b^2q/w;q)_{\infty}}\\ &\qquad\cdot W(b^2/w;b,bc/a,bd/a,be/a,bf/w,bg/w,bh/w;q)\\ &+\frac{(b/a,aq,bf/w,bg/w,bh/w,wq/f,wq/g,wq/h;q)_{\infty}}{(aq/f,aq/g,aq/h,bf/a,bg/a,bh/a,b/w,wq;q)_{\infty}}\\ &\qquad\cdot W(w;b,wc/a,wd/a,we/a,f,g,h;q)\\ \end{align}
となって定理が示される.