Wei-Yuによる8ψ8の変換公式

Baileyの三項変換公式
\begin{align} &\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,aq/de,aq/df,aq/ef,eq/c,fq/c,b/a,bef/a;q)_{\infty}}{(aq/d,aq/e,aq/f,aq/def,q/c,efq/c,be/a,bf/a;q)_{\infty}}\\ &\qquad\cdot\Q87{ef/c,\sqrt{\frac{ef}c}q,-\sqrt{\frac{ef}c}q,aq/bc,aq/cd,ef/a,e,f}{\sqrt{\frac{ef}c},-\sqrt{\frac{ef}c},bef/a,def/a,aq/c,fq/c,eq/c}{\frac{bd}a}\\ &\qquad+\frac{(aq,bq/a,bq/c,bq/d,bq/e,bq/f,d,e,f,aq/bc,bdef/a^2q,a^2q^2/bdef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,bd/a,be/a,bf/a,def/aq,aq^2/def,q/c,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,b,bc/a,bd/a,be/a,bf/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq/a,bq/c,bq/d,bq/e,bq/f}{\frac{a^2q^2}{bcdef}} \end{align}
の1つ目の項に ${}_8\phi_7$の二項変換公式
\begin{align} \Q87{a,\sqrt{a}q,-\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,aq/ef,wq/e,wq/f;q)_{\infty}}{(aq/e,aq/f,wq,wq/ef;q)_{\infty}}\Q87{w,\sqrt wq,-\sqrt wq,wb/a,wc/a,wd/a,e,f}{\sqrt w,-\sqrt w,aq/b,aq/c,aq/d,wq/e,wq/f}{\frac{aq}{ef}}\qquad w=a^2q/bcd \end{align}
を用いると,
\begin{align} &\Q87{ef/c,\sqrt{\frac{ef}c}q,-\sqrt{\frac{ef}c}q,aq/bc,aq/cd,ef/a,e,f}{\sqrt{\frac{ef}c},-\sqrt{\frac{ef}c},bef/a,def/a,aq/c,fq/c,eq/c}{\frac{bd}a}\\ &=\frac{(efq/c,q/c,bdf/a,bde/a;q)_{\infty}}{(fq/c,eq/c,bdef/a,bd/a;q)_{\infty}}\Q87{bdef/aq,\sqrt{\frac{bdef}{aq}}q,-\sqrt{\frac{bdef}{aq}}q,d,b,bcdef/a^2q,e,f}{\sqrt{\frac{bdef}{aq}},-\sqrt{\frac{bdef}{aq}},bef/a,def/a,aq/c,bdf/a,bde/a}{\frac qc} \end{align}
であるからこれを代入して,
\begin{align} &\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,aq/de,aq/df,aq/ef,eq/c,fq/c,b/a,bef/a;q)_{\infty}}{(aq/d,aq/e,aq/f,aq/def,q/c,efq/c,be/a,bf/a;q)_{\infty}}\\ &\qquad\cdot\frac{(efq/c,q/c,bdf/a,bde/a;q)_{\infty}}{(fq/c,eq/c,bdef/a,bd/a;q)_{\infty}}\Q87{bdef/aq,\sqrt{\frac{bdef}{aq}}q,-\sqrt{\frac{bdef}{aq}}q,d,b,bcdef/a^2q,e,f}{\sqrt{\frac{bdef}{aq}},-\sqrt{\frac{bdef}{aq}},bef/a,def/a,aq/c,bdf/a,bde/a}{\frac qc}\\ &\qquad+\frac{(aq,bq/a,bq/c,bq/d,bq/e,bq/f,d,e,f,aq/bc,bdef/a^2q,a^2q^2/bdef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,bd/a,be/a,bf/a,def/aq,aq^2/def,q/c,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,b,bc/a,bd/a,be/a,bf/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq/a,bq/c,bq/d,bq/e,bq/f}{\frac{a^2q^2}{bcdef}}\\ &=\frac{(aq,aq/de,aq/df,aq/ef,eq/c,fq/c,b/a,bef/a,efq/c,q/c,bdf/a,bde/a;q)_{\infty}}{(aq/d,aq/e,aq/f,aq/def,q/c,efq/c,bd/a,be/a,bf/a,fq/c,eq/c,bdef/a;q)_{\infty}}\\ &\qquad\cdot\Q87{bdef/aq,\sqrt{\frac{bdef}{aq}}q,-\sqrt{\frac{bdef}{aq}}q,d,b,bcdef/a^2q,e,f}{\sqrt{\frac{bdef}{aq}},-\sqrt{\frac{bdef}{aq}},bef/a,def/a,aq/c,bdf/a,bde/a}{\frac qc}\\ &\qquad+\frac{(aq,bq/a,bq/c,bq/d,bq/e,bq/f,d,e,f,aq/bc,bdef/a^2q,a^2q^2/bdef;q)_{\infty}}{(aq/b,aq/c,aq/d,aq/e,aq/f,bd/a,be/a,bf/a,def/aq,aq^2/def,q/c,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,b,bc/a,bd/a,be/a,bf/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq/a,bq/c,bq/d,bq/e,bq/f}{\frac{a^2q^2}{bcdef}} \end{align}
$c$と$f$を入れ替えると示すべき等式が得られる.

$N$を非負整数として, 補題1において$a\mapsto aq^{-2N},b\mapsto bq^{-N}, c\mapsto cq^{-N},d\mapsto dq^{-N},e\mapsto eq^{-N},f\mapsto fq^{-N}$とすると, $\mu=bcde/aq$とするとき,
\begin{align} &\Q87{aq^{-2N},\sqrt aq^{1-N},-\sqrt aq^{1-N},bq^{-N},cq^{-N},dq^{-N},eq^{-N},fq^{-N}}{\sqrt aq^{-N},-\sqrt aq^{-N},aq^{1-N}/b,aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,aq^{1-N}/f}{\frac{a^2q^{N+2}}{bcdef}}\\ &=\frac{(aq^{1-N},aq/cd,aq/ce,aq/de,bq^N/a,\mu q^{1-N}/c,\mu q^{1-N}/d,\mu q^{1-N}/e;q)_{\infty}}{(aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,bc/a,bd/a,be/a,bq^N/\mu,\mu q^{1-2N};q)_{\infty}}\\ &\qquad\cdot\Q87{\mu q^{-2N},\sqrt{\mu}q^{1-N},-\sqrt{\mu}q^{1-N},\mu fq^{-N}/a,bq^{-N},cq^{-N},dq^{-N},eq^{-N}}{\sqrt{\mu}q^{-N},-\sqrt{\mu}q^{-N},aq^{1-N}/f,\mu q^{1-N}/b,\mu q^{1-N}/c,\mu q^{1-N}/d,\mu q^{1-N}/e}{\frac{q^{N+1}}f}\\ &\qquad+\frac{(aq^{1-2N},bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f,cq^{-N},dq^{-N},eq^{-N},aq/bf,bcde/a^2q,a^2q^2/bcde;q)_{\infty}}{(aq^{1-N}/b,aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,aq^{1-N}/f,bc/a,bd/a,be/a,cdeq^{-N-1}/a,aq^{N+1}/cde,q^{N+1}/f,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,bq^{-N},bc/a,bd/a,be/a,bf/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f}{\frac{a^2q^{N+2}}{bcdef}} \end{align}
が成り立つ. これは
\begin{align} &\frac{(aq^{-2N},\sqrt aq^{1-N},-\sqrt aq^{1-N},bq^{-N},cq^{-N},dq^{-N},eq^{-N},fq^{-N};q)_N}{(q,\sqrt aq^{-N},-\sqrt aq^{-N},aq^{1-N}/b,aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,aq^{1-N}/f;q)_N}\left(\frac{a^2q^{N+2}}{bcdef}\right)^N\\ &\qquad\cdot\BQ88{aq^{-N},\sqrt aq,-\sqrt aq,b,c,d,e,f}{q^{N+1},\sqrt a,-\sqrt a,aq/b,aq/c,aq/d,aq/e,aq/f}{\frac{a^2q^{N+2}}{bcdef}}\\ &=\frac{(aq^{1-2N},aq/cd,aq/ce,aq/de,bq^N/a,\mu q^{1-N}/c,\mu q^{1-N}/d,\mu q^{1-N}/e;q)_{\infty}}{(aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,bc/a,bd/a,be/a,bq^N/\mu,\mu q^{1-2N};q)_{\infty}}\\ &\qquad\cdot\frac{(\mu q^{-2N},\sqrt{\mu}q^{1-N},-\sqrt{\mu}q^{1-N},\mu fq^{-N}/a,bq^{-N},cq^{-N},dq^{-N},eq^{-N};q)_N}{(q,\sqrt{\mu}q^{-N},-\sqrt{\mu}q^{-N},aq^{1-N}/f,\mu q^{1-N}/b,\mu q^{1-N}/c,\mu q^{1-N}/d,\mu q^{1-N}/e;q)_N}\left(\frac{q^{N+1}}f\right)^N\\ &\qquad\cdot\BQ88{\mu q^{-N},\sqrt{\mu}q,-\sqrt{\mu}q,\mu f/a,b,c,d,e}{\sqrt{\mu},-\sqrt{\mu},aq/f,\mu q/b,\mu q/c,\mu q/d,\mu q/e}{\frac{q^{N+1}}f}\\ &\qquad+\frac{(aq^{1-2N},bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f,cq^{-N},dq^{-N},eq^{-N},aq/bf,bcde/a^2q,a^2q^2/bcde;q)_{\infty}}{(aq^{1-N}/b,aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,aq^{1-N}/f,bc/a,bd/a,be/a,cdeq^{-N-1}/a,aq^{N+2}/cde,q^{N+1}/f,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,bq^{-N},bc/a,bd/a,be/a,bf/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f}{\frac{a^2q^{N+2}}{bcdef}} \end{align}
と書き換えられるので, 整理すると,

\begin{align} &\BQ88{aq^{-N},\sqrt aq,-\sqrt aq,b,c,d,e,f}{q^{N+1},\sqrt a,-\sqrt a,aq/b,aq/c,aq/d,aq/e,aq/f}{\frac{a^2q^{N+2}}{bcdef}}\\ &=\frac{(q,\sqrt aq^{-N},-\sqrt aq^{-N},aq^{1-N}/b,aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,aq^{1-N}/f;q)_N}{(aq^{-2N},\sqrt aq^{1-N},-\sqrt aq^{1-N},bq^{-N},cq^{-N},dq^{-N},eq^{-N},fq^{-N};q)_N}\left(\frac{a^2q^{N+2}}{bcdef}\right)^{-N}\\ &\qquad\cdot\frac{(aq^{1-2N},aq/cd,aq/ce,aq/de,bq^N/a,\mu q^{1-N}/c,\mu q^{1-N}/d,\mu q^{1-N}/e;q)_{\infty}}{(aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,bc/a,bd/a,be/a,bq^N/\mu,\mu q^{1-2N};q)_{\infty}}\\ &\qquad\cdot\frac{(\mu q^{-2N},\sqrt{\mu}q^{1-N},-\sqrt{\mu}q^{1-N},\mu fq^{-N}/a,bq^{-N},cq^{-N},dq^{-N},eq^{-N};q)_N}{(q,\sqrt{\mu}q^{-N},-\sqrt{\mu}q^{-N},aq^{1-N}/f,\mu q^{1-N}/b,\mu q^{1-N}/c,\mu q^{1-N}/d,\mu q^{1-N}/e;q)_N}\left(\frac{q^{N+1}}f\right)^N\\ &\qquad\cdot\BQ88{\mu q^{-N},\sqrt{\mu}q,-\sqrt{\mu}q,\mu f/a,b,c,d,e}{\sqrt{\mu},-\sqrt{\mu},aq/f,\mu q/b,\mu q/c,\mu q/d,\mu q/e}{\frac{q^{N+1}}f}\\ &\qquad\cdot\frac{(q,\sqrt aq^{-N},-\sqrt aq^{-N},aq^{1-N}/b,aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,aq^{1-N}/f;q)_N}{(aq^{-2N},\sqrt aq^{1-N},-\sqrt aq^{1-N},bq^{-N},cq^{-N},dq^{-N},eq^{-N},fq^{-N};q)_N}\left(\frac{a^2q^{N+2}}{bcdef}\right)^{-N}\\ &\qquad+\frac{(aq^{1-2N},bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f,cq^{-N},dq^{-N},eq^{-N},aq/bf,bcde/a^2q,a^2q^2/bcde;q)_{\infty}}{(aq^{1-N}/b,aq^{1-N}/c,aq^{1-N}/d,aq^{1-N}/e,aq^{1-N}/f,bc/a,bd/a,be/a,cdeq^{-N-1}/a,aq^{N+2}/cde,q^{N+1}/f,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,bq^{-N},bc/a,bd/a,be/a,bf/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f}{\frac{a^2q^{N+2}}{bcdef}}\\ &=\frac{(aq,q/a,aq/cd,aq/ce,aq/de,q^{N+1}/f,b/a,\mu q/c,\mu q/d,\mu q/e,aq/f\mu,q^{N+1}/\mu;q)_{\infty}}{(q/f,q^{N+1}/a,aq/c,aq/d,aq/e,bc/a,bd/a,be/a,b/\mu,\mu q,q/\mu,aq^{N+1}/\mu f;q)_{\infty}}\\ &\qquad\cdot\BQ88{\mu q^{-N},\sqrt{\mu}q,-\sqrt{\mu}q,\mu f/a,b,c,d,e}{\sqrt{\mu},-\sqrt{\mu},aq/f,\mu q/b,\mu q/c,\mu q/d,\mu q/e}{\frac{q^{N+1}}f}\\ &\qquad+\frac{(q,aq,q/a,bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f,aq/bf,q^{N+1}/b,bcde/a^2q,a^2q^2/bcde;q)_{\infty}}{(q/f,q^{N+1}/a,aq/b,aq/c,aq/d,aq/e,aq/f,q^{N+1},bc/a,bd/a,be/a,cde/aq,aq^{2}/cde,q/b,b^2q/a;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,bq^{-N},bc/a,bd/a,be/a,bf/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq^{N+1}/a,bq/c,bq/d,bq/e,bq/f}{\frac{a^2q^{N+2}}{bcdef}} \end{align}
と書き換えられる. これは定理1の等式
\begin{align} &\BQ88{\sqrt aq,-\sqrt aq,b,c,d,e,f,g}{\sqrt a,-\sqrt a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g}{\frac{a^3q^2}{bcdefg}}\\ &=\frac{(aq,q/a,aq/cd,aq/ce,aq/de,aq/fg,b/a,\mu q/c,\mu q/d,\mu q/e,aq/\mu f,aq/\mu g;q)_{\infty}}{(q/f,q/g,aq/c,aq/d,aq/e,bc/a,bd/a,be/a,b/\mu,\mu q,q/\mu,a^2q/\mu fg;q)_{\infty}}\\ &\qquad\cdot\BQ88{\sqrt{\mu}q,-\sqrt{\mu}q,b,c,d,e,\mu f/a,\mu g/a}{\sqrt{\mu},-\sqrt{\mu},\mu q/b,\mu q/c,\mu q/e,aq/f,aq/g}{\frac{aq}{fg}}\\ &\qquad+\frac{(q,aq,q/a,c,d,e,bq/c,bq/d,bq/e,bq/f,bq/g,aq/bf,aq/bg,bcde/a^2q,a^2q^2/bcde;q)_{\infty}}{(q/f,q/g,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,bc/a,bd/a,be/a,q/b,b^2q/a,cde/aq,a^2q/cde;q)_{\infty}}\\ &\qquad\cdot\Q87{b^2/a,\sqrt{\frac{b^2}a}q,-\sqrt{\frac{b^2}a}q,bcd/a,bd/a,be/a,bf/a,bg/a}{\sqrt{\frac{b^2}a},-\sqrt{\frac{b^2}a},bq/c,bq/d,bq/e,bq/f,bq/g}{\frac{a^3q^2}{bcdefg}} \end{align}
が$aq/g=q^{N+1}$の場合に成り立つことを意味している. この各項は$aq/g$に関して原点において正則であるから, 一致の定理より示すべき定理が得られる.