Baileyによる8φ7の3項変換公式

Non-terminating $q$-Whippleの変換公式の$q$積分表示より,
\begin{align} &W\left(a;b,c,d,e,f;\frac{a^2q^2}{bcdef}\right)\\ &=\frac{1}{def-aq}\frac{(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/a,q;q)_{\infty}}\int_{aq}^{def}\frac{(atq/bdef,atq/cdef,tq/def,t/a;q)_{\infty}}{(t/de,t/df,t/ef,atq/bcdef;q)_{\infty}}\,d_qt\\ &=\frac{1}{def-aq}\frac{(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/a,q;q)_{\infty}}\int_{aq}^{bdef/a}\frac{(atq/bdef,atq/cdef,tq/def,t/a;q)_{\infty}}{(t/de,t/df,t/ef,atq/bcdef;q)_{\infty}}\,d_qt\\ &\qquad +\frac{1}{def-aq}\frac{(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/a,q;q)_{\infty}}\int_{bdef/a}^{def}\frac{(atq/bdef,atq/cdef,tq/def,t/a;q)_{\infty}}{(t/de,t/df,t/ef,atq/bcdef;q)_{\infty}}\,d_qt\\ \end{align}
であり, 前回の記事 で示した$q$積分
\begin{align} &\int_{a}^b\frac{(tq/a,tq/b,ct,dt;q)_{\infty}}{(et,ft,gt,ht;q)_{\infty}}\,d_qt\\ &=(b-a)\frac{(bc,bd,cd/eh,cd/fh,cd/gh,aq/b,bq/a,q;q)_{\infty}}{(bcd/h,be,bf,bg,bh,ae,af,ag;q)_{\infty}}W\left(bcd/hq;c/h,d/h,be,bf,bg;ah\right)\qquad (cd=abefgh) \end{align}
を用いると, $h$を$aq/bcdef$に選んで,
\begin{align} &\int_{aq}^{bdef/a}\frac{(atq/bdef,atq/cdef,tq/def,t/a;q)_{\infty}}{(t/de,t/df,t/ef,atq/bcdef;q)_{\infty}}\,d_qt\\\\ &=\left(\frac{bdef}a-aq\right)\frac{(bq/a,bq/c,bq/d,bq/e,bq/f,a^2q^2/bdef,bdef/a^2,q;q)_{\infty}}{(b^2q/a,bd/a,be/a,bf/a,q/c,aq/de,aq/df,aq/ef;q)_{\infty}}W\left(b^2/a;b,bc/a,bd/a,be/a,bf/a;\frac{a^2q^2}{bcdef}\right) \end{align}
となる. $h$を$1/ef$に選んで,
\begin{align} &\int_{bdef/a}^{def}\frac{(atq/bdef,atq/cdef,tq/def,t/a;q)_{\infty}}{(t/de,t/df,t/ef,atq/bcdef;q)_{\infty}}\,d_qt\\ &=\left(def-\frac{bdef}a\right)\frac{(aq/c,def/a,eq/c,fq/c,bef/a,bq/a,aq/b,q;q)_{\infty}}{(efq/c,d,e,f,aq/bc,be/a,bf/a,q/c;q)_{\infty}}W\left(ef/c;ef/a,aq/cd,aq/bc,e,f;\frac{bd}{a}\right) \end{align}
だから,
\begin{align} &W\left(a;b,c,d,e,f;\frac{a^2q^2}{bcdef}\right)\\ &=\frac{1}{def-aq}\frac{(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/a,q;q)_{\infty}}\int_{aq}^{bdef/a}\frac{(atq/bdef,atq/cdef,tq/def,t/a;q)_{\infty}}{(t/de,t/df,t/ef,atq/bcdef;q)_{\infty}}\,d_qt\\ &\qquad +\frac{1}{def-aq}\frac{(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/a,q;q)_{\infty}}\int_{bdef/a}^{def}\frac{(atq/bdef,atq/cdef,tq/def,t/a;q)_{\infty}}{(t/de,t/df,t/ef,atq/bcdef;q)_{\infty}}\,d_qt\\ &=\frac{1}{def-aq}\frac{(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/a,q;q)_{\infty}}\\ &\qquad\cdot\,\left(\frac{bdef}a-aq\right)\frac{(bq/a,bq/c,bq/d,bq/e,bq/f,a^2q^2/bdef,bdef/a^2,q;q)_{\infty}}{(b^2q/a,bd/a,be/a,bf/a,q/c,aq/de,aq/df,aq/ef;q)_{\infty}}W\left(b^2/a;b,bc/a,bd/a,be/a,bf/a;\frac{a^2q^2}{bcdef}\right)\\ &\qquad +\frac{1}{def-aq}\frac{(aq,aq/bc,d,e,f,aq/de,aq/df,aq/ef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/a,q;q)_{\infty}}\\ &\qquad \cdot\,\left(def-\frac{bdef}a\right)\frac{(aq/c,def/a,eq/c,fq/c,bef/a,bq/a,aq/b,q;q)_{\infty}}{(efq/c,d,e,f,aq/bc,be/a,bf/a,q/c;q)_{\infty}}W\left(ef/c;ef/a,aq/cd,aq/bc,e,f;\frac{bd}{a}\right)\\ &=\frac{(aq,aq/de,aq/df,aq/ef,eq/c,fq/c,bef/a,b/a;q)_{\infty}}{(aq/d,aq/e,aq/f,aq/def,efq/c,be/a,bf/a,q/c;q)_{\infty}}W\left(ef/c;ef/a,aq/cd,aq/bc,e,f;\frac{bd}{a}\right)\\ &\qquad +\frac{(aq,aq/bc,d,e,f,bq/a,bq/c,bq/d,bq/e,bq/f,a^2q^2/bdef,bdef/a^2q;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,aq^2/def,def/aq,b^2q/a,bd/a,be/a,bf/a,q/c;q)_{\infty}}\\ &\qquad\qquad\cdot\,W\left(b^2/a;b,bc/a,bd/a,be/a,bf/a;\frac{a^2q^2}{bcdef}\right)\\ \end{align}
と示される.