連続q-Jacobi多項式の積公式

$x:=\cos\theta$として,
\begin{align} r_n(x;a,b,c,d|q):=\Q43{q^{-n},abcdq^{n-1},ae^{i\theta},ae^{-i\theta}}{ab,ac,ad}q \end{align}
とする.　連続$q$-Jacobi多項式に対応するのは$q\mapsto q^2$として, $(a,b,c,d)\mapsto (b,bq,-c,-cq)$における場合である. 今回は連続$q$-Jacobi多項式の積公式
\begin{align} &r_n(x;b,bq,-c,-cq|q^2)r_n(y;b,bq,-c,-cq|q^2)\\ &=\int_{-1}^1K(x,y,z)r_n(z;b,bq,-c,-cq|q^2)\,dz \end{align}
における$K(x,y,z)$の明示式を与えるRahmanの結果について解説する. Singhの二次変換公式 を用いると
\begin{align} &r_n(x;b,bq,-c,-cq|q^2)\\ &=\Q43{q^{-2n},b^2c^2q^{2n},be^{i\theta},be^{-i\theta}}{b^2q,-bc,-bcq}{q^2;q^2}\\ &=\Q43{q^{-n},bcq^n,be^{i\theta},be^{-i\theta}}{bq^{\frac 12},-bq^{\frac 12},-bc}{q}\\ &=r_n(x;b,q^{\frac 12},-q^{\frac 12},-c|q)\\ &=\frac{(-cq^{\frac 12},-q;q)_n}{(-bq^{\frac 12},-bc;q)_n}(bq^{-\frac 12})^nr_n(x;q^{\frac 12},b,-c,q^{\frac 12}|q) \end{align}
となる. ここで, 最後の等号は Askey-Wilson多項式の対称性 による. よって, 右辺に現れる多項式
\begin{align} r_n(x;q^{\frac 12},b,-c,q^{\frac 12}|q) \end{align}
の積公式を考えればよい.　 前の記事 で示したAskey-Wilson多項式の積公式を$a,b,c,d$に関して少し入れ替えて$r_n$について書き直した式
\begin{align} &r_n(x;a,b,c,d|q)r_n(y;a,b,c,d|q)\\ &=\frac{(bc,bd;q)_n}{(ac,ad;q)_n}\left(\frac ab\right)^{n}\\ &\qquad\cdot\sum_{k=0}^n\frac{(q^{-n},abcdq^{n-1},ae^{i\theta},ae^{-i\theta},ae^{i\phi},ae^{-i\phi};q)_k}{(q,ab,ab,ac,ad,a/b;q)_k}q^k\\ &\qquad\cdot W(bq^{-k}/a;q^{1-k}/ac,q^{1-k}/ad,q^{-k},be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi};cdq/ab) \end{align}
から始める. ここで, $x=\cos\theta,y=\cos\phi$
\begin{align} W(a;b_1,\dots,b_r;x):=\Q{r+3}{r+2}{a,\sqrt aq,-\sqrt aq,b_1,\dots,b_r}{\sqrt a,-\sqrt a,aq/b_1,\dots,aq/b_r}{x} \end{align}
である. $(a,b,c,d)\mapsto (q^{\frac 12},b,-c,-q^{\frac 12})$とすると
\begin{align} &r_n(x;q^{\frac 12},b,-c,-q^{\frac 12}|q)r_n(y;q^{\frac 12},b,-c,-q^{\frac 12}|q)\\ &=\frac{(-bc,-bq^{\frac 12};q)_n}{(-q,-cq^{\frac 12};q)_n}\left(\frac {q^{\frac 12}}b\right)^{n}\\ &\qquad\cdot\sum_{k=0}^n\frac{(q^{-n},bcq^{n},q^{\frac 12}e^{i\theta},q^{\frac 12}e^{-i\theta},q^{\frac 12}e^{i\phi},q^{\frac 12}e^{-i\phi};q)_k}{(q,bq^{\frac 12},bq^{\frac 12},-cq^{\frac 12},-q,q^{\frac 12}/b;q)_k}q^k\\ &\qquad\cdot W(bq^{-k-\frac 12};-q^{\frac 12-k}/c,-q^{-k},q^{-k},be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi};cq/b) \end{align}
ここで, Rogersの${}_6\phi_5$和公式より
\begin{align} &W(bq^{-k-\frac 12};-q^{\frac 12-k}/c,-q^{-k},q^{-k},be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi};cq/b)\\ &=\sum_{0\leq l}\frac{(1-bq^{2l-k-\frac 12})(bq^{-k-\frac 12},-q^{\frac 12-k}/b,be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_{l}}{(1-bq^{-k-\frac 12})(q,-b^2,e^{-i\theta}q^{\frac 12-k},e^{i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-bq^{\frac 12},bq^{\frac 12};q)_{l}}q^{l}\\ &\qquad\cdot \frac{(-b^2,-q^{\frac 12-k}/c;,q)_l}{(-bc,-q^{\frac 12-k}/b;q)_l}\left(\frac cb\right)^l\\ &=\sum_{0\leq l}\frac{(1-bq^{2l-k-\frac 12})(bq^{-k-\frac 12},-q^{\frac 12-k}/b,be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_{l}}{(1-bq^{-k-\frac 12})(q,-b^2,e^{-i\theta}q^{\frac 12-k},e^{i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-bq^{\frac 12},bq^{\frac 12};q)_{l}}q^{l}\\ &\qquad\cdot \sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,bq^{-k-\frac 12+l},q^{-l};q)_j}{(1+b^2/q)(q,-bc,-bq^{\frac 12-l+k},-b^2q^l;q)_j}\left(-cq^{k+\frac 12}\right)^j\\ &=\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c;q)_j}{(1+b^2/q)(q,-bc;q)_j}\left(\frac{c}{b}\right)^j\\ &\qquad\cdot \sum_{0\leq l}\frac{(1-bq^{2l-k-\frac 12})(bq^{-k-\frac 12};q)_{j+l}(-q^{\frac 12-k}/b;q)_{l-j}(be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_{l}}{(1-bq^{-k-\frac 12})(q;q)_{l-j}(-b^2;q)_{j+l}(e^{-i\theta}q^{\frac 12-k},e^{i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-bq^{\frac 12},bq^{\frac 12};q)_{l}}q^{l}\\ &=\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_j(bq^{\frac 12-k};q)_{2j}}{(1+b^2/q)(q,-bc,e^{-i\theta}q^{\frac 12-k},e^{i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-bq^{\frac 12},bq^{\frac 12};q)_j(-b^2;q)_{2j}}\left(\frac{cq}{b}\right)^j\\ &\qquad\cdot \sum_{0\leq l}\frac{(1-bq^{2j+2l-k-\frac 12})(bq^{2j-k-\frac 12},-q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,be^{-i\theta}q^j,be^{i\phi}q^j,be^{-i\phi}q^j;q)_l}{(1-bq^{2j-k-\frac 12})(q,-b^2q^{2j},-bq^{j+\frac 12},bq^{j+\frac 12},e^{-i\theta}q^{j+\frac 12-k},e^{i\theta}q^{j+\frac 12-k},e^{-i\phi}q^{j+\frac 12-k},e^{i\phi}q^{j+\frac 12-k};q)_l}q^l\\ &=\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_j(bq^{\frac 12-k};q)_{2j}}{(1+b^2/q)(q,-bc,e^{-i\theta}q^{\frac 12-k},e^{i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-bq^{\frac 12},bq^{\frac 12};q)_j(-b^2;q)_{2j}}\left(\frac{cq}{b}\right)^j\\ &\qquad\cdot W(bq^{2j-k-\frac 12};-q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,be^{-i\theta}q^j,be^{i\phi}q^j,be^{-i\phi}q^j;q) \end{align}
と変形できる. ここで右辺の$W$はbalancedであるから, Baileyの${}_{10}\phi_9$変換公式 より,
\begin{align} &W(bq^{2j-k-\frac 12};-q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,be^{-i\theta}q^j,be^{i\phi}q^j,be^{-i\phi}q^j;q)\\ &=\frac{(bq^{2j-k+\frac 12},e^{-i(\theta+\phi)}q^{\frac 12-k}/b,-e^{i\theta}q^{j-k+\frac 12},-e^{i\phi}q^{j-k+\frac 12};q)_{k-j}}{(-be^{i(\theta+\phi)}q^{2j-k+\frac 12},-q^{\frac 12-k}/b,e^{-i\theta}q^{j-k+\frac 12},e^{-i\phi}q^{j-k+\frac 12};q)_{k-j}}\\ &\qquad\cdot W(-be^{i(\theta+\phi)}q^{2j-k-\frac 12};e^{i(\theta+\phi)}q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,-be^{i\theta}q^j,be^{i\phi}q^j,-be^{i\phi}q^j;q)\\ \end{align}
からこれを代入すると,
\begin{align} &W(bq^{-k-\frac 12};-q^{\frac 12-k}/c,-q^{-k},q^{-k},be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi};cq/b)\\ &=\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_j(bq^{\frac 12-k};q)_{2j}}{(1+b^2/q)(q,-bc,e^{-i\theta}q^{\frac 12-k},e^{i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-bq^{\frac 12},bq^{\frac 12};q)_j(-b^2;q)_{2j}}\left(\frac{cq}{b}\right)^j\\ &\qquad\cdot \frac{(bq^{2j-k+\frac 12},e^{-i(\theta+\phi)}q^{\frac 12-k}/b,-e^{i\theta}q^{j-k+\frac 12},-e^{i\phi}q^{j-k+\frac 12};q)_{k-j}}{(-be^{i(\theta+\phi)}q^{2j-k+\frac 12},-q^{\frac 12-k}/b,e^{-i\theta}q^{j-k+\frac 12},e^{-i\phi}q^{j-k+\frac 12};q)_{k-j}}\\ &\qquad\cdot W(-be^{i(\theta+\phi)}q^{2j-k-\frac 12};e^{i(\theta+\phi)}q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,-be^{i\theta}q^j,be^{i\phi}q^j,-be^{i\phi}q^j;q)\\ &=\frac{(-e^{i\theta}q^{\frac 12-k},-e^{i\phi}q^{\frac 12-k};q)_k}{(e^{-i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k};q)_k}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_j}{(1+b^2/q)(q,-bc,e^{i\theta}q^{\frac 12-k},-e^{i\theta}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-e^{i\phi}q^{\frac 12-k},-bq^{\frac 12},bq^{\frac 12};q)_j}\left(\frac{cq}{b}\right)^j\\ &\qquad\cdot \frac{(e^{-i(\theta+\phi)}q^{\frac 12-k}/b;q)_{k-j}}{(-q^{\frac 12-k}/b;q)_{k-j}}\frac{(-be^{i(\theta+\phi)}q^{\frac 12-k};q)_{2j}(bq^{\frac 12-k};q)_{j+k}}{(-b^2;q)_{2j}(-be^{i(\theta+\phi)}q^{\frac 12-k};q)_{j+k}}\\ &\qquad\cdot W(-be^{i(\theta+\phi)}q^{2j-k-\frac 12};e^{i(\theta+\phi)}q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,-be^{i\theta}q^j,be^{i\phi}q^j,-be^{i\phi}q^j;q)\\ &=\frac{(-e^{i\theta}q^{\frac 12-k},-e^{i\phi}q^{\frac 12-k},bq^{\frac 12-k},e^{-i(\theta+\phi)}q^{\frac 12-k}/b;q)_k}{(e^{-i\theta}q^{\frac 12-k},e^{-i\phi}q^{\frac 12-k},-q^{\frac 12-k}/b,-be^{i(\theta+\phi)}q^{\frac 12-k};q)_k}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi},-q^{-k},q^{-k};q)_j}{(1+b^2/q)(q,-bc,e^{i\theta}q^{\frac 12-k},-e^{i\theta}q^{\frac 12-k},e^{i\phi}q^{\frac 12-k},-e^{i\phi}q^{\frac 12-k},-be^{i(\theta+\phi)}q^{\frac 12},-be^{i(\theta+\phi)}q^{\frac 12};q)_j}\left(-\frac{ce^{i(\theta+\phi)}q}{b}\right)^j\\ &\qquad\cdot \frac{(-be^{i(\theta+\phi)}q^{\frac 12-k};q)_{2j}}{(-b^2;q)_{2j}}\\ &\qquad\cdot W(-be^{i(\theta+\phi)}q^{2j-k-\frac 12};e^{i(\theta+\phi)}q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,-be^{i\theta}q^j,be^{i\phi}q^j,-be^{i\phi}q^j;q)\\ &=\frac{(-e^{-i\theta}q^{\frac 12},-e^{-i\phi}q^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12};q)_k}{(e^{i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},-bq^{\frac 12},-e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{-i\theta},be^{-i\phi};q)_j}{(1+b^2/q)(q,-bc,-be^{i\theta},-be^{i\phi};q)_j}\left(-\frac{ce^{i(\theta+\phi)}q}{b}\right)^j\\ &\qquad\cdot \frac{(b^2e^{2i\theta},b^2e^{2i\phi},q^{-2k};q^2)_j}{(e^{2i\theta}q^{1-2k},e^{2i\phi}q^{1-2k},b^2e^{2i(\theta+\phi)}q;q^2)_j}\frac{(-be^{i(\theta+\phi)}q^{\frac 12-k};q)_{2j}}{(-b^2;q)_{2j}}\\ &\qquad\cdot W(-be^{i(\theta+\phi)}q^{2j-k-\frac 12};e^{i(\theta+\phi)}q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,-be^{i\theta}q^j,be^{i\phi}q^j,-be^{i\phi}q^j;q)\\ \end{align}
ここで, Verma-Jainの変換公式( 前の記事 の系1)より,
\begin{align} &W(-be^{i(\theta+\phi)}q^{2j-k-\frac 12};e^{i(\theta+\phi)}q^{\frac 12-k}/b,-q^{j-k},q^{j-k},be^{i\theta}q^j,-be^{i\theta}q^j,be^{i\phi}q^j,-be^{i\phi}q^j;q)\\ &=\frac{(b^2e^{2i(\theta+\phi)}q^{4j-2k+1},q^{1-2k}/b^2;q^2)_{k-j}}{(e^{2i\theta}q^{2j-2k+1},e^{2i\phi}q^{2j-2k+1};q^2)_{k-j}}\Q43{b^2e^{2i\theta}q^{2j},b^2e^{2i\phi}q^{2j},b^2q^{2j},q^{2j-2k}}{b^4q^{4j},be^{i(\theta+\phi)}q^{2j-k+\frac 12},be^{i(\theta+\phi)}q^{2j-k+\frac 32}}{q^2;q^2} \end{align}
であるから, これを代入すると,
\begin{align} &W(bq^{-k-\frac 12};-q^{\frac 12-k}/c,-q^{-k},q^{-k},be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi};cq/b)\\ &=\frac{(-e^{-i\theta}q^{\frac 12},-e^{-i\phi}q^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12};q)_k}{(e^{i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},-bq^{\frac 12},-e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{-i\theta},be^{-i\phi};q)_j}{(1+b^2/q)(q,-bc,-be^{i\theta},-be^{i\phi};q)_j}\left(-\frac{ce^{i(\theta+\phi)}q}{b}\right)^j\\ &\qquad\cdot \frac{(b^2e^{2i\theta},b^2e^{2i\phi},q^{-2k};q^2)_j}{(e^{2i\theta}q^{1-2k},e^{2i\phi}q^{1-2k},b^2e^{2i(\theta+\phi)}q;q^2)_j}\frac{(-be^{i(\theta+\phi)}q^{\frac 12-k};q)_{2j}}{(-b^2;q)_{2j}}\\ &\qquad\cdot \frac{(b^2e^{2i(\theta+\phi)}q^{4j-2k+1},q^{1-2k}/b^2;q^2)_{k-j}}{(e^{2i\theta}q^{2j-2k+1},e^{2i\phi}q^{2j-2k+1};q^2)_{k-j}}\sum_{0\leq l}\frac{(b^2e^{2i\theta}q^{2j},b^2e^{2i\phi}q^{2j},b^2q^{2j},q^{2j-2k};q^2)_l}{(q^2,b^4q^{4j},be^{i(\theta+\phi)}q^{2j-k+\frac 12},be^{i(\theta+\phi)}q^{2j-k+\frac 32};q^2)_l}q^{2l}\\ &=\frac{(-e^{-i\theta}q^{\frac 12},-e^{-i\phi}q^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12};q)_k}{(e^{i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},-bq^{\frac 12},-e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k(e^{2i\theta}q^{1-2k},e^{2i\phi}q^{1-2k};q^2)_k}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{-i\theta},be^{-i\phi};q)_j}{(1+b^2/q)(q,-bc,-be^{i\theta},-be^{i\phi};q)_j}\left(-\frac{ce^{i(\theta+\phi)}q}{b}\right)^j\\ &\qquad\cdot \frac{(b^4;q^2)_{2j}(be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_j}{(b^2,b^2e^{2i(\theta+\phi)}q;q^2)_j}\frac{(-be^{i(\theta+\phi)}q^{\frac 12-k};q)_{2j}}{(-b^2;q)_{2j}}\frac{(b^2e^{2i(\theta+\phi)}q^{1-2k};q^2)_{j+k}(q^{1-2k}/b^2;q^2)_{k-j}}{(b^2e^{2i(\theta+\phi)}q^{1-2k};q^2)_{2j}}\\ &\qquad\cdot \sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{j+l}}{(q^2;q^2)_l(b^4;q^2)_{l+2j}(be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{j+l}}q^{2l}\\ &=\frac{(-e^{-i\theta}q^{\frac 12},-e^{-i\phi}q^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12};q)_k}{(e^{i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},-bq^{\frac 12},-e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k(e^{2i\theta}q^{1-2k},e^{2i\phi}q^{1-2k};q^2)_k}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{-i\theta},be^{-i\phi};q)_j}{(1+b^2/q)(q,-bc,-be^{i\theta},-be^{i\phi};q)_j}\left(-\frac{ce^{i(\theta+\phi)}}{bq}\right)^j\\ &\qquad\cdot \frac{(b^2q;q^2)_{j}(b^2e^{2i(\theta+\phi)}q^{1-2k};q^2)_{j+k}(q^{1-2k}/b^2;q^2)_{k-j}}{(b^2e^{2i(\theta+\phi)}q;q^2)_j}\\ &\qquad\cdot \sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2;q^2)_{l-j}(b^4;q^2)_{l+j}(be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &=\frac{(-e^{-i\theta}q^{\frac 12},-e^{-i\phi}q^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12};q)_k(b^2e^{2i(\theta+\phi)}q^{1-2k},q^{1-2k}/b^2;q^2)_k}{(e^{i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},-bq^{\frac 12},-e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k(e^{2i\theta}q^{1-2k},e^{2i\phi}q^{1-2k};q^2)_k}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{-i\theta},be^{-i\phi};q)_j}{(1+b^2/q)(q,-bc,-be^{i\theta},-be^{i\phi};q)_j}\left(bce^{i(\theta+\phi)}\right)^jq^{2\binom j2}\\ &\qquad\cdot \sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2;q^2)_{l-j}(b^4;q^2)_{l+j}(be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &=\frac{(-e^{-i\theta}q^{\frac 12},-e^{-i\phi}q^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12};q)_k(e^{-2i(\theta+\phi)}q/b^2,b^2q;q^2)_k}{(e^{i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},-bq^{\frac 12},-e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k(e^{-2i\theta}q,e^{-2i\phi}q;q^2)_k}\\ &\qquad\cdot \sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^4,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\sum_{0\leq j}\frac{(1+b^2q^{2j-1})(-b^2/q,b/c,be^{-i\theta},be^{-i\phi};q)_j(q^{-2l};q^2)_j}{(1+b^2/q)(q,-bc,-be^{i\theta},-be^{i\phi};q)_j(b^4q^{2l};q^2)_j}\left(-bce^{i(\theta+\phi)}q^{2l}\right)^j\\ &=\frac{(bq^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}{(e^{i\theta}q^{\frac 12},e^{-i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},e^{-i\phi}q^{\frac 12};q)_k}\sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^4,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot W(-b^2/q;b/c,be^{-i\theta},be^{-i\phi},-q^{-l},q^{-l};-bce^{i(\theta+\phi)}q^{2l}) \end{align}
と変形できる. ここで, ${}_8\phi_7$の二次変換公式 より,
\begin{align} &W(-b^2/q;b/c,be^{-i\theta},be^{-i\phi},-q^{-l},q^{-l};-bce^{i(\theta+\phi)}q^{2l})\\ &=\frac{(-b^2,-bce^{i(\theta+\phi)}q^l;q)_l}{(b^2q^l,bce^{i(\theta+\phi)};q)_l}W(bce^{i(\theta+\phi)}/q;-e^{i(\theta+\phi)},-ce^{i\theta},-ce^{i\phi},-q^{-l},q^{-l};b^2q^{2l})\\ &=\frac{(b^4;q^2)_l(-bce^{i(\theta+\phi)};q)_{2l}}{(b^2c^2e^{2i(\theta+\phi)};q^2)_l(b^2;q)_{2l}}W(bce^{i(\theta+\phi)}/q;-e^{i(\theta+\phi)},-ce^{i\theta},-ce^{i\phi},-q^{-l},q^{-l};b^2q^{2l}) \end{align}
であるから, これを代入して,
\begin{align} &W(bq^{-k-\frac 12};-q^{\frac 12-k}/c,-q^{-k},q^{-k},be^{i\theta},be^{-i\theta},be^{i\phi},be^{-i\phi};cq/b)\\ &=\frac{(bq^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}{(e^{i\theta}q^{\frac 12},e^{-i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},e^{-i\phi}q^{\frac 12};q)_k}\sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^4,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\frac{(b^4;q^2)_l(-bce^{i(\theta+\phi)};q)_{2l}}{(b^2c^2e^{2i(\theta+\phi)};q^2)_l(b^2;q)_{2l}}W(bce^{i(\theta+\phi)}/q;-e^{i(\theta+\phi)},-ce^{i\theta},-ce^{i\phi},-q^{-l},q^{-l};b^2q^{2l})\\ &=\frac{(bq^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}{(e^{i\theta}q^{\frac 12},e^{-i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},e^{-i\phi}q^{\frac 12};q)_k}\sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^2c^2e^{2i(\theta+\phi)},be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\frac{(-bce^{i(\theta+\phi)};q)_{2l}}{(b^2;q)_{2l}}W(bce^{i(\theta+\phi)}/q;-e^{i(\theta+\phi)},-ce^{i\theta},-ce^{i\phi},-q^{-l},q^{-l};b^2q^{2l}) \end{align}
を得る. これより,
\begin{align} &r_n(x;q^{\frac 12},b,-c,-q^{\frac 12}|q)r_n(y;q^{\frac 12},b,-c,-q^{\frac 12}|q)\\ &=\frac{(-bc,-bq^{\frac 12};q)_n}{(-q,-cq^{\frac 12};q)_n}\left(\frac {q^{\frac 12}}b\right)^{n}\\ &\qquad\cdot\sum_{k=0}^n\frac{(q^{-n},bcq^{n},q^{\frac 12}e^{i\theta},q^{\frac 12}e^{-i\theta},q^{\frac 12}e^{i\phi},q^{\frac 12}e^{-i\phi};q)_k}{(q,bq^{\frac 12},bq^{\frac 12},-cq^{\frac 12},-q,q^{\frac 12}/b;q)_k}q^k\\ &\qquad\cdot \frac{(bq^{\frac 12},q^{\frac 12}/b,be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}{(e^{i\theta}q^{\frac 12},e^{-i\theta}q^{\frac 12},e^{i\phi}q^{\frac 12},e^{-i\phi}q^{\frac 12};q)_k}\sum_{0\leq l}\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^2c^2e^{2i(\theta+\phi)},be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\frac{(-bce^{i(\theta+\phi)};q)_{2l}}{(b^2;q)_{2l}}W(bce^{i(\theta+\phi)}/q;-e^{i(\theta+\phi)},-ce^{i\theta},-ce^{i\phi},-q^{-l},q^{-l};b^2q^{2l})\\ &=\frac{(-bc,-bq^{\frac 12};q)_n}{(-q,-cq^{\frac 12};q)_n}\left(\frac {q^{\frac 12}}b\right)^{n}\\ &\qquad\cdot\sum_{k=0}^n\frac{(q^{-n},bcq^{n},be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}{(q,bq^{\frac 12},-cq^{\frac 12},-q;q)_k}q^k\\ &\qquad\cdot \sum_{l=0}^k\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^2c^2e^{2i(\theta+\phi)},be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\frac{(-bce^{i(\theta+\phi)};q)_{2l}}{(b^2;q)_{2l}}W(bce^{i(\theta+\phi)}/q;-e^{i(\theta+\phi)},-ce^{i\theta},-ce^{i\phi},-q^{-l},q^{-l};b^2q^{2l}) \end{align}
となる. ここで, Nassrallah-Rahman積分( 前の記事 の定理2)より
\begin{align} &W(bce^{i(\theta+\phi)}/q;-e^{i(\theta+\phi)},-ce^{i\theta},-ce^{i\phi},-q^{-l},q^{-l};b^2q^{2l})\\ &=\frac{(q,be^{-i\theta},be^{-i\phi},bcq^l,-bcq^l,b/c,be^{i\theta}q^l,-be^{i\theta}q^l,be^{i\phi}q^l,-be^{i\phi}q^l,-bce^{i(\theta+\phi)}q^{2l},bce^{i(\theta+\phi)};q)_{\infty}}{2\pi(-bc,-be^{i\theta},-be^{-i\phi},-bce^{i(\theta+\phi)}q^l,bce^{i(\theta+\phi)}q^l,b^2q^{2l};q)_{\infty}}\\ &\qquad\cdot \int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi},\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^l,-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^{l};q)_{\infty}}\right|^2\,d\psi\\ &=\frac{(q,bc,b/c;q)_{\infty}|(be^{i\theta},be^{i\phi};q)_{\infty}|^2}{2\pi(b^2;q)_{\infty}}\frac{(b^2c^2e^{2i(\theta+\phi)};q^2)_l(b^2;q)_{2l}}{(b^2c^2,b^2e^{2i\theta},b^2e^{2i\phi};q^2)_l(-bce^{i(\theta+\phi)};q)_{2l}}\\ &\qquad\cdot \int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi},\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^l,-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^{l};q)_{\infty}}\right|^2\,d\psi\\ \end{align}
と表されるので, これを代入して,

\begin{align} &r_n(x;q^{\frac 12},b,-c,-q^{\frac 12}|q)r_n(y;q^{\frac 12},b,-c,-q^{\frac 12}|q)\\ &=\frac{(q,bc,b/c;q)_{\infty}|(be^{i\theta},be^{i\phi};q)_{\infty}|^2}{2\pi(b^2;q)_{\infty}}\frac{(-bc,-bq^{\frac 12};q)_n}{(-q,-cq^{\frac 12};q)_n}\left(\frac {q^{\frac 12}}b\right)^{n}\\ &\qquad\cdot\sum_{k=0}^n\frac{(q^{-n},bcq^{n},be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}{(q,bq^{\frac 12},-cq^{\frac 12},-q;q)_k}q^k\\ &\qquad\cdot \sum_{l=0}^k\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^2c^2e^{2i(\theta+\phi)},be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\frac{(-bce^{i(\theta+\phi)};q)_{2l}}{(b^2;q)_{2l}}\frac{(b^2c^2e^{2i(\theta+\phi)};q^2)_l(b^2;q)_{2l}}{(b^2c^2,b^2e^{2i\theta},b^2e^{2i\phi};q^2)_l(-bce^{i(\theta+\phi)};q)_{2l}}\\ &\qquad\cdot \int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi},\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^l,-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^{l};q)_{\infty}}\right|^2\,d\psi \end{align}
ここで,
\begin{align} &\sum_{l=0}^k\frac{(b^2e^{2i\theta},b^2e^{2i\phi},b^2,q^{-2k};q^2)_{l}}{(q^2,b^2c^2e^{2i(\theta+\phi)},be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\frac{(-bce^{i(\theta+\phi)};q)_{2l}}{(b^2;q)_{2l}}\frac{(b^2c^2e^{2i(\theta+\phi)};q^2)_l(b^2;q)_{2l}}{(b^2c^2,b^2e^{2i\theta},b^2e^{2i\phi};q^2)_l(-bce^{i(\theta+\phi)};q)_{2l}}\\ &\qquad\cdot \int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi},\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^l,-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi}q^{l};q)_{\infty}}\right|^2\,d\psi\\ &=\int_0^{\pi}\left|\frac{(e^{2i\psi};q)_{\infty}}{(\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},\sqrt{bc}e^{i(\theta+\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi};q)_{\infty}}\right|^2\,d\psi\\ &\qquad\cdot \sum_{l=0}^k\frac{(b^2,bc e^{i(\theta+\phi)+2i\psi},bc e^{i(\theta+\phi)-2i\psi},q^{-2k};q^2)_{l}}{(q^2,b^2c^2,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l} \end{align}
であり, Askey-Wilson積分 より,
\begin{align} &\frac{(b^2,bce^{i(\theta+\phi)+2i\psi},bce^{i(\theta+\phi)-2i\psi};q^2)_l}{(b^2c^2;q^2)_l}\\ &=\frac{(q^2,b^2,bce^{i(\theta+\phi)+2i\psi},bce^{i(\theta+\phi)-2i\psi},bce^{-i(\theta+\phi)+2i\psi},bce^{-i(\theta+\phi)-2i\psi},c^2;q^2)_{\infty}}{2\pi(b^2c^2;q^2)_{\infty}}\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\omega};q^2)_{\infty}}{(be^{i(\theta+\phi)+i\omega}q^{2l},be^{-i(\theta+\phi)+i\omega},ce^{2i\psi+i\omega},ce^{-2i\psi+i\omega};q^2)_{\infty}}\right|^2\,d\omega \end{align}
であるからこれを代入して,
\begin{align} &\int_0^{\pi}\left|\frac{(e^{2i\psi};q)_{\infty}}{(\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},\sqrt{bc}e^{i(\theta+\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi};q)_{\infty}}\right|^2\,d\psi\\ &\qquad\cdot \sum_{l=0}^k\frac{(b^2,bc e^{i(\theta+\phi)+2i\psi},bc e^{i(\theta+\phi)-2i\psi},q^{-2k};q^2)_{l}}{(q^2,b^2c^2,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &=\int_0^{\pi}\left|\frac{(e^{2i\psi};q)_{\infty}}{(\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},\sqrt{bc}e^{i(\theta+\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi};q)_{\infty}}\right|^2\,d\psi\\ &\qquad\cdot \sum_{l=0}^k\frac{(q^{-2k};q^2)_{l}}{(q^2,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &\qquad\cdot\frac{(q^2,b^2,bce^{i(\theta+\phi)+2i\psi},bce^{i(\theta+\phi)-2i\psi},bce^{-i(\theta+\phi)+2i\psi},bce^{-i(\theta+\phi)-2i\psi},c^2;q^2)_{\infty}}{2\pi(b^2c^2;q^2)_{\infty}}\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\omega};q^2)_{\infty}}{(be^{i(\theta+\phi)+i\omega}q^{2l},be^{-i(\theta+\phi)+i\omega},ce^{2i\psi+i\omega},ce^{-2i\psi+i\omega};q^2)_{\infty}}\right|^2\,d\omega\\ &=\frac{(q^2,b^2,c^2;q^2)_{\infty}}{2\pi(b^2c^2;q^2)_{\infty}}\int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi};q)_{\infty}}\right|^2\,d\psi\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\omega};q^2)_{\infty}}{(be^{i(\theta+\phi)+i\omega},be^{-i(\theta+\phi)+i\omega},ce^{2i\psi+i\omega},ce^{-2i\psi+i\omega};q^2)_{\infty}}\right|^2\,d\omega\\ &\qquad\cdot \sum_{l=0}^k\frac{(be^{i(\theta+\phi)+i\omega},be^{i(\theta+\phi)-i\omega},q^{-2k};q^2)_{l}}{(q^2,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l} \end{align}
ここで, $q$-Saalschützの和公式 より,
\begin{align} &\sum_{l=0}^k\frac{(be^{i(\theta+\phi)+i\omega},be^{i(\theta+\phi)-i\omega},q^{-2k};q^2)_{l}}{(q^2,be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k};q^2)_{l}}q^{2l}\\ &=\frac{(e^{i\omega}q^{\frac 12-k},e^{-i\omega}q^{\frac 12-k};q^2)_k}{(be^{i(\theta+\phi)}q^{\frac 12-k},e^{-i(\theta+\phi)}q^{\frac 12-k}/b;q^2)_k}\\ &=\frac{(e^{i\omega}q^{\frac 12-k},e^{i\omega}q^{\frac 32-k};q^2)_k}{(be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 32-k}/b;q^2)_k}(be^{i(\theta+\phi)})^k\\ &=\frac{(e^{i\omega}q^{\frac 12-k},e^{i\omega}q^{\frac 12};q)_{k}}{(be^{i(\theta+\phi)}q^{\frac 12-k},be^{i(\theta+\phi)}q^{\frac 12};q)_{k}}(be^{i(\theta+\phi)})^k\\ &=\frac{(e^{i\omega}q^{\frac 12},e^{-i\omega}q^{\frac 12};q)_{k}}{(be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_{k}} \end{align}
であるからこれを代入してまとめると,

\begin{align} &r_n(x;q^{\frac 12},b,-c,-q^{\frac 12}|q)r_n(y;q^{\frac 12},b,-c,-q^{\frac 12}|q)\\ &=\frac{(q,bc,b/c;q)_{\infty}|(be^{i\theta},be^{i\phi};q)_{\infty}|^2}{2\pi(b^2;q)_{\infty}}\frac{(-bc,-bq^{\frac 12};q)_n}{(-q,-cq^{\frac 12};q)_n}\left(\frac {q^{\frac 12}}b\right)^{n}\\ &\qquad\cdot\sum_{k=0}^n\frac{(q^{-n},bcq^{n},be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_k}{(q,bq^{\frac 12},-cq^{\frac 12},-q;q)_k}q^k\\ &\qquad\cdot \frac{(q^2,b^2,c^2;q^2)_{\infty}}{2\pi(b^2c^2;q^2)_{\infty}}\int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi};q)_{\infty}}\right|^2\,d\psi\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\omega};q^2)_{\infty}}{(be^{i(\theta+\phi)+i\omega},be^{-i(\theta+\phi)+i\omega},ce^{2i\psi+i\omega},ce^{-2i\psi+i\omega};q^2)_{\infty}}\right|^2\,d\omega\\ &\qquad\cdot \frac{(e^{i\omega}q^{\frac 12},e^{-i\omega}q^{\frac 12};q)_{k}}{(be^{i(\theta+\phi)}q^{\frac 12},e^{-i(\theta+\phi)}q^{\frac 12}/b;q)_{k}}\\ &=\frac{(q,bc,b/c;q)_{\infty}(q^2,b^2,c^2;q^2)_{\infty}|(be^{i\theta},be^{i\phi};q)_{\infty}|^2}{4\pi^2(b^2;q)_{\infty}(b^2c^2;q^2)_{\infty}}\frac{(-bc,-bq^{\frac 12};q)_n}{(-q,-cq^{\frac 12};q)_n}\left(\frac {q^{\frac 12}}b\right)^{n}\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi};q)_{\infty}}\right|^2\,d\psi\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\omega};q^2)_{\infty}}{(be^{i(\theta+\phi)+i\omega},be^{-i(\theta+\phi)+i\omega},ce^{2i\psi+i\omega},ce^{-2i\psi+i\omega};q^2)_{\infty}}\right|^2\,d\omega\\ &\qquad\cdot r_n(\cos\omega;q^{\frac 12},b,-c,-q^{\frac 12}|q) \end{align}
最初に示した式
\begin{align} r_n(x;b,bq,-c,-cq|q^2)&=\frac{(-cq^{\frac 12},-q;q)_n}{(-bq^{\frac 12},-bc;q)_n}(bq^{-\frac 12})^nr_n(x;q^{\frac 12},b,-c,q^{\frac 12}|q) \end{align}
を思い出すとこれは,
\begin{align} &r_n(x;b,bq,-c,-cq|q^2)r_n(y;b,bq,-c,-cq|q^2)\\ &=\frac{(q,bc,b/c;q)_{\infty}(q^2,b^2,c^2;q^2)_{\infty}|(be^{i\theta},be^{i\phi};q)_{\infty}|^2}{4\pi^2(b^2;q)_{\infty}(b^2c^2;q^2)_{\infty}}\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\psi},-\sqrt{bc}e^{-i(\theta+\phi)/2+i\psi},-\sqrt{bc}e^{i(\theta+\phi)/2+i\psi};q)_{\infty}}{(\sqrt{\frac bc}e^{i(\theta-\phi)/2+i\psi},\sqrt{\frac bc}e^{i(\phi-\theta)/2+i\psi};q)_{\infty}}\right|^2\,d\psi\\ &\qquad\cdot\int_0^{\pi}\left|\frac{(e^{2i\omega};q^2)_{\infty}}{(be^{i(\theta+\phi)+i\omega},be^{-i(\theta+\phi)+i\omega},ce^{2i\psi+i\omega},ce^{-2i\psi+i\omega};q^2)_{\infty}}\right|^2\,d\omega\\ &\qquad\cdot r_n(\cos\omega;b,bq,-c,-cq|q^2) \end{align}
と書き換えられる. $\psi,\omega$を入れ替えて, 以下が得られた.

$K(x,y,z)$に現れる積分は
\begin{align} &\int_0^{\pi}\left|\frac{(e^{2i\omega},-\sqrt{bc}e^{-i(\theta+\phi)+i\omega},-\sqrt{bc}e^{i(\theta+\phi)+i\omega};q)_{\infty}}{(\sqrt{\frac bc}e^{i(\theta-\phi)+i\omega},\sqrt{\frac bc}e^{i(\phi-\theta)+i\omega};q)_{\infty}(ce^{i\psi+2i\omega},ce^{-i\psi+2i\omega};q^2)}\right|^2\,d\omega\\ &=\int_0^{\pi}\left|\frac{(e^{2i\omega},-\sqrt{bc}e^{-i(\theta+\phi)+i\omega},-\sqrt{bc}e^{i(\theta+\phi)+i\omega};q)_{\infty}}{(\sqrt{\frac bc}e^{i(\theta-\phi)+i\omega},\sqrt{\frac bc}e^{i(\phi-\theta)+i\omega},\sqrt ce^{i\psi/2+i\omega},-\sqrt ce^{i\psi/2+i\omega},\sqrt ce^{-i\psi/2+i\omega},-\sqrt ce^{-i\psi/2+i\omega};q)_{\infty}}\right|^2\,d\omega \end{align}
と書き換えられるので, Rahmanの超幾何積分 を用いて${}_{10}\phi_9$に書き換えることもできる.