Non-terminating q-Whippleの変換公式

$w=a^2q/bcd, a^3q^{n+2}=bcdefg$とする. Baileyのterminating${}_{10}\phi_9$変換公式
\begin{align} &\Q{10}9{a,\sqrt aq,-\sqrt aq,b,c,d,e,f,g,q^{-n}}{\sqrt a,-\sqrt a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq^{n+1}}{q}\\ &=\frac{(aq,aq/ef,wq/e,wq/f;q)_n}{(aq/e,aq/f,wq,wq/ef;q)_n}\Q{10}9{w,\sqrt wq,-\sqrt wq,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,wq^{n+1}}{q} \end{align}
において, $a,c,d,e,f,g$を固定して$n\to \infty$とする. $b=a^3q^{n+2}/cdefg, w=efgq^{-n-1}/a$となるから,
\begin{align} &\Q87{a,\sqrt aq,-\sqrt aq,c,d,e,f,g}{\sqrt a,-\sqrt a,aq/c,aq/d,aq/e,aq/f,aq/g}{\frac{a^2q^2}{cdefg}}\\ &=\lim_{n\to\infty}\frac{(aq,aq/ef,fgq^{-n}/a,egq^{-n}/a;q)_n}{(aq/e,aq/f,gq^{-n}/a,efgq^{-n}/a)_n}\Q{10}9{w,\sqrt wq,-\sqrt wq,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,wq^{n+1}}{q}\\ &=\frac{(aq,aq/ef,aq/eg,aq/fg;q)_{\infty}}{(aq/e,aq/f,aq/g,aq/efg;q)_{\infty}}\lim_{n\to\infty}\Q{10}9{w,\sqrt wq,-\sqrt wq,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,wq^{n+1}}{q} \end{align}
$n$を奇数として, $n=2m+1$とすると,
\begin{align} &=\lim_{n\to\infty}\Q{10}9{w,\sqrt wq,-\sqrt wq,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,wq^{n+1}}{q}\\ &=\lim_{n\to\infty}\sum_{k=0}^m\frac{(w,\sqrt wq,-\sqrt wq,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,wq^{n+1},q;q)_k}q^k\\ &\qquad +\lim_{n\to\infty}\sum_{k=0}^m\frac{(w,\sqrt wq,-\sqrt wq,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,wq^{n+1},q;q)_{n-k}}q^{n-k}\\ \end{align}
最初の項は,
\begin{align} &\lim_{n\to\infty}\sum_{k=0}^m\frac{(w,\sqrt wq,-\sqrt wq,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,wq^{n+1},q;q)_k}q^k\\ &=\lim_{n\to\infty}\sum_{k=0}^m\frac{(efgq^{-n-1}/a,\sqrt{w}q,-\sqrt {w}q,aq/cd,cefgq^{-n-1}/a^2,defgq^{-n-1}/a^2,e,f,g,q^{-n};q)_k}{(\sqrt{w},-\sqrt w,cdefgq^{-n-1}/a^2,aq/c,aq/d,fgq^{-n}/a,egq^{-n}/a,efq^{-n}/a,efg/a,q;q)_k}q^k\\ &=\sum_{k=0}^{\infty}\frac{(aq/cd,e,f,g;q)_k}{(aq/c,aq/d,efg/a,q;q)_k}q^k \end{align}
であり, 次の項は,
\begin{align} &\lim_{n\to\infty}\sum_{k=0}^m\frac{(w,\sqrt wq,-\sqrt wq,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,wq^{n+1},q;q)_{n-k}}q^{n-k}\\ &\lim_{n\to\infty}\frac{(w,\sqrt wq,-\sqrt wq,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,wq^{n+1},q;q)_n}q^n\\ &\qquad \cdot \sum_{k=0}^m\frac{(q^{1-n}/\sqrt w,-q^{1-n}/\sqrt w, bq^{-n}/a,cq^{-n}/a,dq^{-n}/a,eq^{-n}/w,fq^{-n}/w,gq^{-n}/w,q^{-2n}/w,q^{-n};q)_k}{(q^{1-n}/w,q^{-n}/\sqrt w,-q^{-n}/\sqrt w,aq^{1-n}/wb,aq^{1-n}/wc,aq^{1-n}/wd,q^{1-n}/e,q^{1-n}/f,q^{1-n}/g,q;q)_k}q^k \end{align}
係数の部分は,
\begin{align} &\lim_{n\to\infty}\frac{(efgq^{-n-1}/a,\sqrt wq,-\sqrt wq,aq/cd,cefgq^{-n-1}/a^2,defgq^{-n-1}/a^2,e,f,g,q^{-n};q)_n}{(\sqrt w,-\sqrt w,cdefgq^{-n-1}/a^2,aq/c,aq/d,fgq^{-n}/a,egq^{-n}/a,efq^{-n}/a,efg/a,q;q)_n}q^n\\ &=\lim_{n\to\infty}\frac{1-efgq^{n-1}/a}{q^n-efg/aq}\frac{(aq^2/efg,aq/cd,a^2q^2/cefg,a^2q^2/defg,e,f,g;q)_n}{(a^2q^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/a;q)_n}\\ &=-\frac{aq}{efg}\frac{(aq^2/efg,aq/cd,a^2q^2/cefg,a^2q^2/defg,e,f,g;q)_{\infty}}{(a^2q^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/a;q)_{\infty}}\\ &=\frac{(aq/efg,aq/cd,a^2q^2/cefg,a^2q^2/defg,e,f,g;q)_{\infty}}{(a^2q^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/aq;q)_{\infty}} \end{align}
であり,
\begin{align} &\lim_{n\to\infty}\sum_{k=0}^m\frac{(q^{1-n}/\sqrt w,-q^{1-n}/\sqrt w, bq^{-n}/a,cq^{-n}/a,dq^{-n}/a,eq^{-n}/w,fq^{-n}/w,gq^{-n}/w,q^{-2n}/w,q^{-n};q)_k}{(q^{1-n}/w,q^{-n}/\sqrt w,-q^{-n}/\sqrt w,aq^{1-n}/wb,aq^{1-n}/wc,aq^{1-n}/wd,q^{1-n}/e,q^{1-n}/f,q^{1-n}/g,q;q)_k}q^k\\ &=\lim_{n\to\infty}\sum_{k=0}^m\frac{(q^{1-n}/\sqrt w,-q^{1-n}/\sqrt w, a^2q^2/cdefg,cq^{-n}/a,dq^{-n}/a,aq/fg,aq/eg,aq/ef,aq^{1-n}/efg,q^{-n};q)_k}{(aq^2/efg,q^{-n}/\sqrt w,-q^{-n}/\sqrt w,cdq^{-n}/a,a^2q^{2}/cefg,a^2q^{2}/defg,q^{1-n}/e,q^{1-n}/f,q^{1-n}/g,q;q)_k}q^k\\ &=\sum_{k=0}^{\infty}\frac{(a^2q^2/cdefg,aq/fg,aq/eg,aq/ef;q)_k}{(aq^2/efg,a^2q^{2}/cefg,a^2q^{2}/defg,q;q)_k}q^k\\ \end{align}
よってこれらを合わせると,
\begin{align} &\Q87{a,\sqrt aq,-\sqrt aq,c,d,e,f,g}{\sqrt a,-\sqrt a,aq/c,aq/d,aq/e,aq/f,aq/g}{\frac{a^2q^2}{cdefg}}\\ &=\frac{(aq,aq/ef,aq/eg,aq/fg;q)_{\infty}}{(aq/e,aq/f,aq/g,aq/efg;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(aq/cd,e,f,g;q)_k}{(aq/c,aq/d,efg/a,q;q)_k}q^k\\ &\qquad +\frac{(aq,aq/ef,aq/eg,aq/fg;q)_{\infty}}{(aq/e,aq/f,aq/g,aq/efg;q)_{\infty}}\frac{(aq/efg,aq/cd,a^2q^2/cefg,a^2q^2/defg,e,f,g;q)_{\infty}}{(a^2q^2/cdefg,aq/c,aq/d,aq/fg,aq/eg,aq/ef,efg/aq;q)_{\infty}}\\ &\qquad\qquad\cdot\sum_{k=0}^{\infty}\frac{(a^2q^2/cdefg,aq/fg,aq/eg,aq/ef;q)_k}{(aq^2/efg,a^2q^{2}/cefg,a^2q^{2}/defg,q;q)_k}q^k\\ &=\frac{(aq,aq/ef,aq/eg,aq/fg;q)_{\infty}}{(aq/e,aq/f,aq/g,aq/efg;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(aq/cd,e,f,g;q)_k}{(aq/c,aq/d,efg/a,q;q)_k}q^k\\ &\qquad +\frac{(aq,aq/cd,a^2q^2/cefg,a^2q^2/defg,e,f,g;q)_{\infty}}{(aq/c,aq/d,aq/e,aq/f,aq/g,a^2q^2/cdefg,efg/aq;q)_{\infty}}\\ &\qquad\qquad\cdot\sum_{k=0}^{\infty}\frac{(a^2q^2/cdefg,aq/fg,aq/eg,aq/ef;q)_k}{(aq^2/efg,a^2q^{2}/cefg,a^2q^{2}/defg,q;q)_k}q^k\\ \end{align}
となる. 文字を置き換えることによって定理を得る.