Gasper-RahmanによるSears-Carlitzの公式の拡張

BaileyのNearly-Poised${}_5\phi_4$の変換公式
\begin{align} &\Q54{a,b,c,d,q^{-n}}{aq/b,aq/c,aq/d,a^2q^{-n}/w^2}{q}\\ &=\frac{(wq/a,w^2q/a;q)_n}{(wq,w^2q/a^2;q)_n}\Q{12}{11}{w,\sqrt wq,-\sqrt wq,wb/a,wc/a,wd/a,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},w^2q^{n+1}/a,q^{-n}}{\sqrt w,-\sqrt w,aq/b,aq/c,aq/d,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},aq^{-n}/w,wq^{n+1}}q\qquad(w=a^2q/bcd) \end{align}
において, $d\mapsto dq^n$とすると, $w=a^2q^{1-n}/bcd$であり,
\begin{align} &\Q54{a,b,c,dq^n,q^{-n}}{aq/b,aq/c,aq^{1-n}/d,b^2c^2d^2q^{n-2}/a^2}q\\ &=\frac{(aq^{2-n}/bcd,a^3q^{3-2n}/b^2c^2d^2;q)_n}{(a^2q^{2-n}/bcd,a^2q^{3-2n}/b^2c^2d^2;q)_n}\Q{12}{11}{w,\sqrt wq,-\sqrt wq,aq^{1-n}/cd,aq^{1-n}/bd,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},a^3q^{3-n}/b^2c^2d^2,q^{-n}}{\sqrt w,-\sqrt w,aq/b,aq/c,aq^{1-n}/d,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},bcd/aq,a^2q^2/bcd}q \end{align}
よって, $n\to\infty$とすると,
\begin{align} &\Q32{a,b,c}{aq/b,aq/c}{\frac{d}a}\\ &=\lim_{n\to\infty}\frac{(aq^{2-n}/bcd,a^3q^{3-2n}/b^2c^2d^2;q)_n}{(a^2q^{2-n}/bcd,a^2q^{3-2n}/b^2c^2d^2;q)_n}\Q{12}{11}{w,\sqrt wq,-\sqrt wq,aq^{1-n}/cd,aq^{1-n}/bd,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},a^3q^{3-n}/b^2c^2d^2,q^{-n}}{\sqrt w,-\sqrt w,aq/b,aq/c,aq/d,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},bcd/aq,a^2q^2/bcd}q\\ &=\frac{(bcd/aq;q)_{\infty}}{(bcd/a^2q;q)_{\infty}}\lim_{n\to\infty}\Q{12}{11}{w,\sqrt wq,-\sqrt wq,aq^{1-n}/cd,aq^{1-n}/bd,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},a^3q^{3-n}/b^2c^2d^2,q^{-n}}{\sqrt w,-\sqrt w,aq/b,aq/c,aq^{1-n}/d,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},bcd/aq,a^2q^2/bcd}q \end{align}
ここで, $d=a^2qx/bc$とすると, $w=q^{-n}/x$であり,
\begin{align} &\Q32{a,b,c}{aq/b,aq/c}{\frac{aqx}{bc}}\\ &=\frac{(ax;q)_{\infty}}{(x;q)_{\infty}}\lim_{n\to\infty}\Q{12}{11}{w,\sqrt wq,-\sqrt wq,bq^{-n}/ax,cq^{-n}/ax,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},q^{1-n}/ax^2,q^{-n}}{\sqrt w,-\sqrt w,aq/b,aq/c,bcq^{-n}/ax,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},ax,q/x}q \end{align}
$n$を奇数として, $n=2m+1$とすると,
\begin{align} &\lim_{n\to\infty}\Q{12}{11}{w,\sqrt wq,-\sqrt wq,bq^{-n}/ax,cq^{-n}/ax,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},q^{1-n}/ax^2,q^{-n}}{\sqrt w,-\sqrt w,aq/b,aq/c,bcq^{-n}/ax,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},ax,q/x}q\\ &=\lim_{n\to\infty}\sum_{k=0}^m\frac{(w,\sqrt wq,-\sqrt wq,bq^{-n}/ax,cq^{-n}/ax,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},q^{1-n}/ax^2,q^{-n};q)_k}{(\sqrt w,-\sqrt w,aq/b,aq/c,bcq^{-n}/ax,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},ax,q/x,q;q)_k}q^k\\ &\qquad+\lim_{n\to\infty}\sum_{k=0}^m\frac{(w,\sqrt wq,-\sqrt wq,bq^{-n}/ax,cq^{-n}/ax,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},q^{1-n}/ax^2,q^{-n};q)_{n-k}}{(\sqrt w,-\sqrt w,aq/b,aq/c,bcq^{-n}/ax,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},ax,q/x,q;q)_{n-k}}q^{n-k}\\ &=\sum_{k=0}^{\infty}\frac{(aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq};q)_k}{(aq/b,aq/c,ax,q/x,q;q)_k}q^k\\ &\qquad+\lim_{n\to\infty}\frac{(w,\sqrt wq,-\sqrt wq,bq^{-n}/ax,cq^{-n}/ax,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},q^{1-n}/ax^2,q^{-n};q)_n}{(\sqrt w,-\sqrt w,aq/b,aq/c,bcq^{-n}/ax,wq/\sqrt a,-wq/\sqrt a,w\sqrt{q/a},-w\sqrt{q/a},ax,q/x,q;q)_n}q^n\\ &\qquad\qquad\cdot\sum_{k=0}^m\frac{(q^{1-n}/\sqrt w,-q^{1-n}/\sqrt w,bq^{-n}/a,cq^{-n}/a,axq/bc,x\sqrt a,-x\sqrt a,x\sqrt{aq},-x\sqrt{aq},q^{1-n}/ax,xq^{-n},q^{-n};q)_k}{(xq,q^{-n}/\sqrt w,-q^{-n}/\sqrt w,axq/b,axq/c,bcq^{-n}/a,q^{1-n}/\sqrt a,-q^{1-n}/\sqrt a,q^{1-n}/\sqrt{aq},-q^{1-n}/\sqrt{aq},ax^2,q;q)_{k}}q^{k}\\ &=\sum_{k=0}^{\infty}\frac{(aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq};q)_k}{(aq/b,aq/c,ax,q/x,q;q)_k}q^k\\ &\qquad-x\lim_{n\to\infty}\frac{(xq,axq/b,axq/c,aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq},ax^2;q)_n}{(aq/b,aq/c,axq/bc,x\sqrt a,-x\sqrt a,x\sqrt{aq},-x\sqrt{aq},ax,q/x,q;q)_n}\\ &\qquad\qquad\cdot\sum_{k=0}^{\infty}\frac{(axq/bc,x\sqrt a,-x\sqrt a,x\sqrt{aq},-x\sqrt{aq};q)_k}{(xq,axq/b,axq/c,ax^2,q;q)_{k}}q^{k}\\ &=\sum_{k=0}^{\infty}\frac{(aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq};q)_k}{(aq/b,aq/c,ax,q/x,q;q)_k}q^k\\ &\qquad-x\frac{(xq,axq/b,axq/c,aq/bc,a;q)_{\infty}}{(aq/b,aq/c,axq/bc,ax,q/x,q;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(axq/bc,x\sqrt a,-x\sqrt a,x\sqrt{aq},-x\sqrt{aq};q)_k}{(xq,axq/b,axq/c,ax^2,q;q)_{k}}q^{k}\\ &=\sum_{k=0}^{\infty}\frac{(aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq};q)_k}{(aq/b,aq/c,ax,q/x,q;q)_k}q^k\\ &\qquad+\frac{(x,axq/b,axq/c,aq/bc,a;q)_{\infty}}{(aq/b,aq/c,axq/bc,ax,1/x,q;q)_{\infty}}\sum_{k=0}^{\infty}\frac{(axq/bc,x\sqrt a,-x\sqrt a,x\sqrt{aq},-x\sqrt{aq};q)_k}{(xq,axq/b,axq/c,ax^2,q;q)_{k}}q^{k}\\ \end{align}
となるから,
\begin{align} &\Q32{a,b,c}{aq/b,aq/c}{\frac{axq}{bc}}\\ &=\frac{(ax;q)_{\infty}}{(x;q)_{\infty}}\Q54{aq/bc,\sqrt a,-\sqrt a,\sqrt{aq},-\sqrt{aq}}{aq/b,aq/c,ax,q/x}q\\ &\qquad+\frac{(a,aq/bc,axq/b,axq/c;q)_{\infty}}{(aq/b,aq/c,axq/bc,1/x;q)_{\infty}}\Q54{axq/bc,x\sqrt a,-x\sqrt a,x\sqrt{aq},-x\sqrt{aq}}{axq/b,axq/c,xq,ax^2}q \end{align}
を得る.