Non-terminating q-Whippleの変換公式

$w=a^{2}q/bcd,a^3{q}^{n+2}=bcdefg$とする. Baileyのterminating${}_{10}{\varphi }_9$変換公式
$$\begin{array}{rl}& {}_{10}{\varphi }_9[\begin{array}{c}a,\sqrt{a}q,-\sqrt{a}q,b,c,d,e,f,g,q^{-n}\\ \sqrt{a},-\sqrt{a},aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,a{q}^{n+1}\end{array};q]\\ & =\frac{(aq,aq/ef,wq/e,wq/f;q{)}_n}{(aq/e,aq/f,wq,wq/ef;q{)}_n}{}_{10}{\varphi }_9[\begin{array}{c}w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n}\\ \sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1}\end{array};q]\end{array}$$
において, $a,c,d,e,f,g$を固定して$n\to \mathrm{\infty }$とする. $b=a^3{q}^{n+2}/cdefg,w=efg{q}^{-n-1}/a$となるから,
$$\begin{array}{rl}& {}_8{\varphi }_7[\begin{array}{c}a,\sqrt{a}q,-\sqrt{a}q,c,d,e,f,g\\ \sqrt{a},-\sqrt{a},aq/c,aq/d,aq/e,aq/f,aq/g\end{array};\frac{a^2{q}^2}{cdefg}]\\ & =\underset{n\to \mathrm{\infty }}{lim}\frac{(aq,aq/ef,fg{q}^{-n}/a,eg{q}^{-n}/a;q{)}_n}{(aq/e,aq/f,g{q}^{-n}/a,efg{q}^{-n}/a{)}_n}{}_{10}{\varphi }_9[\begin{array}{c}w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n}\\ \sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1}\end{array};q]\\ & =\frac{(aq,aq/ef,aq/eg,aq/fg;q{)}_{\mathrm{\infty }}}{(aq/e,aq/f,aq/g,aq/efg;q{)}_{\mathrm{\infty }}}\underset{n\to \mathrm{\infty }}{lim}{}_{10}{\varphi }_9[\begin{array}{c}w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n}\\ \sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1}\end{array};q]\end{array}$$
$n$を奇数として, $n=2m+1$とすると,
$$\begin{array}{rl}& =\underset{n\to \mathrm{\infty }}{lim}{}_{10}{\varphi }_9[\begin{array}{c}w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n}\\ \sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1}\end{array};q]\\ & =\underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^m\frac{(w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n};q{)}_k}{(\sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1},q;q{)}_k}{q}^k\\ & +\underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^m\frac{(w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n};q{)}_{n-k}}{(\sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1},q;q{)}_{n-k}}{q}^{n-k}\end{array}$$
最初の項は,
$$\begin{array}{rl}& \underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^m\frac{(w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n};q{)}_k}{(\sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1},q;q{)}_k}{q}^k\\ & =\underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^m\frac{(efg{q}^{-n-1}/a,\sqrt{w}q,-\sqrt{w}q,aq/cd,cefg{q}^{-n-1}/a^2,defg{q}^{-n-1}/a^2,e,f,g,q^{-n};q{)}_k}{(\sqrt{w},-\sqrt{w},cdefg{q}^{-n-1}/a^2,aq/c,aq/d,fg{q}^{-n}/a,eg{q}^{-n}/a,ef{q}^{-n}/a,efg/a,q;q{)}_k}{q}^k\\ & =\sum _{k=0}^{\mathrm{\infty }}\frac{(aq/cd,e,f,g;q{)}_k}{(aq/c,aq/d,efg/a,q;q{)}_k}{q}^k\end{array}$$
であり, 次の項は,
$$\begin{array}{rl}& \underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^m\frac{(w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n};q{)}_{n-k}}{(\sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1},q;q{)}_{n-k}}{q}^{n-k}\\ & \underset{n\to \mathrm{\infty }}{lim}\frac{(w,\sqrt{w}q,-\sqrt{w}q,wb/a,wc/a,wd/a,e,f,g,q^{-n};q{)}_n}{(\sqrt{w},-\sqrt{w},aq/b,aq/c,aq/d,wq/e,wq/f,wq/g,w{q}^{n+1},q;q{)}_n}{q}^n\\ & \cdot \sum _{k=0}^m\frac{(q^{1-n}/\sqrt{w},-q^{1-n}/\sqrt{w},b{q}^{-n}/a,c{q}^{-n}/a,d{q}^{-n}/a,e{q}^{-n}/w,f{q}^{-n}/w,g{q}^{-n}/w,q^{-2n}/w,q^{-n};q{)}_k}{(q^{1-n}/w,q^{-n}/\sqrt{w},-q^{-n}/\sqrt{w},a{q}^{1-n}/wb,a{q}^{1-n}/wc,a{q}^{1-n}/wd,q^{1-n}/e,q^{1-n}/f,q^{1-n}/g,q;q{)}_k}{q}^k\end{array}$$
係数の部分は,
$$\begin{array}{rl}& \underset{n\to \mathrm{\infty }}{lim}\frac{(efg{q}^{-n-1}/a,\sqrt{w}q,-\sqrt{w}q,aq/cd,cefg{q}^{-n-1}/a^2,defg{q}^{-n-1}/a^2,e,f,g,q^{-n};q{)}_n}{(\sqrt{w},-\sqrt{w},cdefg{q}^{-n-1}/a^2,aq/c,aq/d,fg{q}^{-n}/a,eg{q}^{-n}/a,ef{q}^{-n}/a,efg/a,q;q{)}_n}{q}^n\\ & =\underset{n\to \mathrm{\infty }}{lim}\frac{1-efg{q}^{n-1}/a}{q^n-efg/aq}\frac{(a{q}^2/efg,aq/cd,a^2{q}^2/cefg,a^2{q}^2/defg,e,f,g;q{)}_n}{(a^2{q}^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/a;q{)}_n}\\ & =-\frac{aq}{efg}\frac{(a{q}^2/efg,aq/cd,a^2{q}^2/cefg,a^2{q}^2/defg,e,f,g;q{)}_{\mathrm{\infty }}}{(a^2{q}^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/a;q{)}_{\mathrm{\infty }}}\\ & =\frac{(aq/efg,aq/cd,a^2{q}^2/cefg,a^2{q}^2/defg,e,f,g;q{)}_{\mathrm{\infty }}}{(a^2{q}^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/aq;q{)}_{\mathrm{\infty }}}\end{array}$$
であり,
$$\begin{array}{rl}& \underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^m\frac{(q^{1-n}/\sqrt{w},-q^{1-n}/\sqrt{w},b{q}^{-n}/a,c{q}^{-n}/a,d{q}^{-n}/a,e{q}^{-n}/w,f{q}^{-n}/w,g{q}^{-n}/w,q^{-2n}/w,q^{-n};q{)}_k}{(q^{1-n}/w,q^{-n}/\sqrt{w},-q^{-n}/\sqrt{w},a{q}^{1-n}/wb,a{q}^{1-n}/wc,a{q}^{1-n}/wd,q^{1-n}/e,q^{1-n}/f,q^{1-n}/g,q;q{)}_k}{q}^k\\ & =\underset{n\to \mathrm{\infty }}{lim}\sum _{k=0}^m\frac{(q^{1-n}/\sqrt{w},-q^{1-n}/\sqrt{w},a^2{q}^2/cdefg,c{q}^{-n}/a,d{q}^{-n}/a,aq/fg,aq/eg,aq/ef,a{q}^{1-n}/efg,q^{-n};q{)}_k}{(a{q}^2/efg,q^{-n}/\sqrt{w},-q^{-n}/\sqrt{w},cd{q}^{-n}/a,a^2{q}^2/cefg,a^2{q}^2/defg,q^{1-n}/e,q^{1-n}/f,q^{1-n}/g,q;q{)}_k}{q}^k\\ & =\sum _{k=0}^{\mathrm{\infty }}\frac{(a^2{q}^2/cdefg,aq/fg,aq/eg,aq/ef;q{)}_k}{(a{q}^2/efg,a^2{q}^2/cefg,a^2{q}^2/defg,q;q{)}_k}{q}^k\end{array}$$
よってこれらを合わせると,
$$\begin{array}{rl}& {}_8{\varphi }_7[\begin{array}{c}a,\sqrt{a}q,-\sqrt{a}q,c,d,e,f,g\\ \sqrt{a},-\sqrt{a},aq/c,aq/d,aq/e,aq/f,aq/g\end{array};\frac{a^2{q}^2}{cdefg}]\\ & =\frac{(aq,aq/ef,aq/eg,aq/fg;q{)}_{\mathrm{\infty }}}{(aq/e,aq/f,aq/g,aq/efg;q{)}_{\mathrm{\infty }}}\sum _{k=0}^{\mathrm{\infty }}\frac{(aq/cd,e,f,g;q{)}_k}{(aq/c,aq/d,efg/a,q;q{)}_k}{q}^k\\ & +\frac{(aq,aq/ef,aq/eg,aq/fg;q{)}_{\mathrm{\infty }}}{(aq/e,aq/f,aq/g,aq/efg;q{)}_{\mathrm{\infty }}}\frac{(aq/efg,aq/cd,a^2{q}^2/cefg,a^2{q}^2/defg,e,f,g;q{)}_{\mathrm{\infty }}}{(a^2{q}^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/aq;q{)}_{\mathrm{\infty }}}\\ & \cdot \sum _{k=0}^{\mathrm{\infty }}\frac{(a^2{q}^2/cdefg,aq/fg,aq/eg,aq/ef;q{)}_k}{(a{q}^2/efg,a^2{q}^2/cefg,a^2{q}^2/defg,q;q{)}_k}{q}^k\\ & =\frac{(aq,aq/ef,aq/eg,aq/fg;q{)}_{\mathrm{\infty }}}{(aq/e,aq/f,aq/g,aq/efg;q{)}_{\mathrm{\infty }}}\sum _{k=0}^{\mathrm{\infty }}\frac{(aq/cd,e,f,g;q{)}_k}{(aq/c,aq/d,efg/a,q;q{)}_k}{q}^k\\ & +\frac{(aq,aq/cd,a^2{q}^2/cefg,a^2{q}^2/defg,e,f,g;q{)}_{\mathrm{\infty }}}{(aq/c,aq/d,aq/e,aq/f,aq/g,a^2{q}^2/cdefg,efg/aq;q{)}_{\mathrm{\infty }}}\\ & \cdot \sum _{k=0}^{\mathrm{\infty }}\frac{(a^2{q}^2/cdefg,aq/fg,aq/eg,aq/ef;q{)}_k}{(a{q}^2/efg,a^2{q}^2/cefg,a^2{q}^2/defg,q;q{)}_k}{q}^k\end{array}$$
となる. 文字を置き換えることによって定理を得る.