Non-terminating Watsonの変換公式の直接的な証明

Rogersの${}_6\phi_5$和公式 より
\begin{align} &\Q43{aq/bc,d,e,f}{aq/b,aq/c,def/a}q\\ &=\sum_{0\leq n}\frac{(d,e,f;q)_n}{(q,aq,def/a;q)_n}q^n\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c,q^{-n};q)_k}{(1-a)(q,aq/b,aq/c,aq^{n+1};q)_k}\left(\frac{aq^{n+1}}{bc}\right)^k\\ &=\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c;q)_k}{(1-a)(q,aq/b,aq/c;q)_k}\left(-\frac{aq}{bc}\right)^kq^{\binom k2}\sum_{0\leq n}\frac{(d,e,f;q)_n}{(def/a;q)_n(aq;q)_{n+k}(q;q)_{n-k}}q^n\\ &=\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c,d,e,f;q)_k}{(1-a)(q,aq/b,aq/c,def/a;q)_k(aq;q)_{2k}}\left(-\frac{aq^2}{bc}\right)^kq^{\binom{k}2}\Q32{dq^k,eq^k,fq^k}{defq^k/a,aq^{2k+1}}q \end{align}
ここで, non-terminating $q$-Saalschützの和公式 より
\begin{align} &\Q32{dq^k,eq^k,fq^k}{defq^k/a,aq^{2k+1}}q\\ &=\frac{(aq^{1-k}/def,aq^{k+1}/d,aq^{k+1}/e,aq^{k+1}/f;q)_{\infty}}{(aq/de,aq/df,aq/ef,aq^{2k+1};q)_{\infty}}\\ &\qquad-\frac{(aq^{1-k}/def,dq^k,eq^k,fq^k,a^2q^{k+2}/def;q)_{\infty}}{(defq^{k-1}/a,aq/de,aq/df,aq/ef,aq^{2k+1};q)_{\infty}}\Q32{aq/de,aq/df,aq/ef}{a^2q^{k+2}/def,aq^{1-k}/def}q\\ &=\frac{(aq/def,aq/d,aq/e,aq/f;q)_{\infty}}{(aq/de,aq/df,aq/ef,aq;q)_{\infty}}\frac{(aq;q)_{2k}(def/a;q)_k}{(aq/d,aq/e,aq/f;q)_k}\left(-\frac{a}{def}\right)^kq^{-\binom k2}\\ &\qquad-\frac{(aq/def,d,e,f,a^2q^{2}/def;q)_{\infty}}{(def/aq,aq/de,aq/df,aq/ef,aq;q)_{\infty}}\frac{(aq;q)_{2k}(def/a,def/aq;q)_k}{(d,e,f,a^2q^2/def;q)_k}\left(-\frac{a}{def}\right)^kq^{-\binom k2}\Q32{aq/de,aq/df,aq/ef}{a^2q^{k+2}/def,aq^{2-k}/def}q \end{align}
であるからこれを代入すると
\begin{align} &\Q43{aq/bc,d,e,f}{aq/b,aq/c,def/a}q\\ &=\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c,d,e,f;q)_k}{(1-a)(q,aq/b,aq/c,def/a;q)_k(aq;q)_{2k}}\left(-\frac{aq^2}{bc}\right)^kq^{\binom{k}2}\\ &\qquad\cdot\frac{(aq/def,aq/d,aq/e,aq/f;q)_{\infty}}{(aq/de,aq/df,aq/ef,aq;q)_{\infty}}\frac{(aq;q)_{2k}(def/a;q)_k}{(aq/d,aq/e,aq/f;q)_k}\left(-\frac{a}{def}\right)^kq^{-\binom k2}\\ &\qquad-\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c,d,e,f;q)_k}{(1-a)(q,aq/b,aq/c,def/a;q)_k(aq;q)_{2k}}\left(-\frac{aq^2}{bc}\right)^kq^{\binom{k}2}\\ &\qquad\cdot\frac{(aq/def,d,e,f,a^2q^{2}/def;q)_{\infty}}{(def/aq,aq/de,aq/df,aq/ef,aq;q)_{\infty}}\frac{(aq;q)_{2k}(def/a,def/aq;q)_k}{(d,e,f,a^2q^2/def;q)_k}\left(-\frac{a}{def}\right)^kq^{-\binom k2}\Q32{aq/de,aq/df,aq/ef}{a^2q^{k+2}/def,aq^{2-k}/def}q\\ &=\frac{(aq/def,aq/d,aq/e,aq/f;q)_{\infty}}{(aq/de,aq/df,aq/ef,aq;q)_{\infty}}\Q87{a,\sqrt aq,-\sqrt aq,b,c,d,e,f}{\sqrt a,-\sqrt a,aq/b,aq/c,aq/d,aq/e,aq/f}{\frac{a^2q^2}{bcdef}}\\ &-\frac{(aq/def,d,e,f,a^2q^{2}/def;q)_{\infty}}{(def/aq,aq/de,aq/df,aq/ef,aq;q)_{\infty}}\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c,def/aq;q)_k}{(1-a)(q,aq/b,aq/c,a^2q^2/def;q)_k}\left(\frac{a^2q^2}{bcdef}\right)^k\\ &\qquad\cdot\sum_{0\leq j}\frac{(aq/de,aq/df,aq/ef;q)_j}{(q,a^2q^{k+2}/def,aq^{2-k}/def;q)_j}q^j \end{align}
ここで, 第二項は Rogersの和公式 より
\begin{align} &\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c,def/aq;q)_k}{(1-a)(q,aq/b,aq/c,a^2q^2/def;q)_k}\left(\frac{a^2q^2}{bcdef}\right)^k\\ &\qquad\cdot\sum_{0\leq j}\frac{(aq/de,aq/df,aq/ef;q)_j}{(q,a^2q^{k+2}/def,aq^{1-k}/def;q)_j}q^j\\ &=\sum_{0\leq j}\frac{(aq/de,aq/df,aq/ef;q)_j}{(q,aq^2/def,a^2q^2/def;q)_j}q^j\sum_{0\leq k}\frac{(1-aq^{2k})(a,b,c,def/aq^{j+1};q)_k}{(1-a)(q,aq/b,aq/c,a^2q^{j+2}/def;q)_k}\left(\frac{a^2q^{j+2}}{bcdef}\right)^k\\ &=\sum_{0\leq j}\frac{(aq/de,aq/df,aq/ef;q)_j}{(q,aq^2/def,a^2q^2/def;q)_j}q^j\frac{(aq,aq/bc,a^2q^{j+2}/bdef,a^2q^{j+2}/cdef;q)_{\infty}}{(aq/b,aq/c,a^2q^{j+2}/def,a^2q^{j+2}/bcdef;q)_{\infty}}\\ &=\frac{(aq,aq/bc,a^2q^2/bdef,a^2q^2/cdef;q)_{\infty}}{(aq/b,aq/c,a^2q^2/def,a^2q^2/bcdef;q)_{\infty}}\Q43{aq/de,aq/df,aq/ef,a^2q^2/bcdef}{aq^2/def,a^2q^2/bdef,a^2q^2/cdef}q \end{align}
であるからこれを代入すると,
\begin{align} &\Q43{aq/bc,d,e,f}{aq/b,aq/c,def/a}q\\ &=\frac{(aq/def,aq/d,aq/e,aq/f;q)_{\infty}}{(aq/de,aq/df,aq/ef,aq;q)_{\infty}}\Q87{a,\sqrt aq,-\sqrt aq,b,c,d,e,f}{\sqrt a,-\sqrt a,aq/b,aq/c,aq/d,aq/e,aq/f}{\frac{a^2q^2}{bcdef}}\\ &-\frac{(aq/def,d,e,f,a^2q^{2}/def;q)_{\infty}}{(def/aq,aq/de,aq/df,aq/ef,aq;q)_{\infty}}\\ &\qquad\cdot\frac{(aq,aq/bc,a^2q^2/bdef,a^2q^2/cdef;q)_{\infty}}{(aq/b,aq/c,a^2q^2/def,a^2q^2/bcdef;q)_{\infty}}\Q43{aq/de,aq/df,aq/ef,a^2q^2/bcdef}{aq^2/def,a^2q^2/bdef,a^2q^2/cdef}q\\ &=\frac{(aq/def,aq/d,aq/e,aq/f;q)_{\infty}}{(aq/de,aq/df,aq/ef,aq;q)_{\infty}}\Q87{a,\sqrt aq,-\sqrt aq,b,c,d,e,f}{\sqrt a,-\sqrt a,aq/b,aq/c,aq/d,aq/e,aq/f}{\frac{a^2q^2}{bcdef}}\\ &-\frac{(aq/def,d,e,f,aq/bc,a^2q^2/bdef,a^2q^2/cdef;q)_{\infty}}{(def/aq,aq/de,aq/df,aq/ef,aq/b,aq/c,a^2q^2/bcdef;q)_{\infty}}\\ &\qquad\cdot\Q43{aq/de,aq/df,aq/ef,a^2q^2/bcdef}{aq^2/def,a^2q^2/bdef,a^2q^2/cdef}q\\ \end{align}
となるので, ${}_8\phi_7$に関して整理して定理を得る.