Baileyによる10φ9の4項変換公式

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