連続q-Jacobi多項式の線形化公式3: F_jの計算

前の記事 で, 連続$q$-Jacobi多項式の線形化公式の係数を以下の$F_j$の計算に帰着した.
\begin{align} F_j&=\sum_{\lambda=0}^{j}\frac{(q^{-j},-bcq^{2s+j},q^{s-n},q^{1+s-n}/bc,-bq^{n+\frac 12},-cq^{n+\frac 12};q)_{\lambda}}{(q,bq^{s+\frac 12},cq^{s+\frac 12},-q^{1+2s-2n}/bc,-1,-bcq^{2n+1};q)_{\lambda}}q^{\lambda}K_{\lambda}\\ K_{\lambda}&=\Q{10}9{-q^{-\lambda},q\sqrt{-q^{\lambda}},-q\sqrt{-q^{\lambda}},-q^{1+s-n}/bc,-q^{s-n},bq^{n+\frac 12},cq^{n+\frac 12},q^{-\lambda-s},-q^{-\lambda-s},q^{-\lambda}}{\sqrt{-q^{\lambda}},-\sqrt{-q^{\lambda}},bcq^{n-\lambda-s},q^{1+n-s-\lambda},-q^{\frac 12-\lambda-n}/b,-q^{\frac 12-\lambda-n}/c,-q^{s+1},q^{s+1},-q}q \end{align}
その 次の記事 で
\begin{align} K_{\lambda} &=\frac{(-1,-q^{1+2s-2n}/bc;q)_{\lambda}}{( q^{1-n+s}/bc,q^{s-n};q)_{\lambda}}\\ &\qquad\cdot\sum_{0\leq r}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,q^{2s-2n},q^{-\lambda},q^{1-\lambda};q^2)_r(cq^{n+\frac 12};q)_r}{(q^{2s+2},q^2;q^2)_r(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/b,-q^{\frac 12-\lambda-n}/c;q)_r(-q^{1+2s-2n}/bc;q)_{2r}}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\Q43{-b/c,q^{2r-\lambda},-q^{1+s-n+r}/bc,-q^{s-n+r}}{-bq^{n+\frac 12+r},-q^{\frac 12-\lambda-n+r}/c,-q^{1+2s-2n+2r}/bc}{q} \end{align}
という表示を導いた. 今回はこれを用いることによって$F_j$の計算を行いたいと思う. 上の表示を代入すると,
\begin{align} F_j&=\sum_{\lambda=0}^{j}\frac{(q^{-j},-bcq^{2s+j},-bq^{n+\frac 12},-cq^{n+\frac 12};q)_{\lambda}}{(q,bq^{s+\frac 12},cq^{s+\frac 12},-bcq^{2n+1};q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\sum_{0\leq r}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,q^{2s-2n},q^{-\lambda},q^{1-\lambda};q^2)_r(cq^{n+\frac 12};q)_r}{(q^{2s+2},q^2;q^2)_r(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/b,-q^{\frac 12-\lambda-n}/c;q)_r(-q^{1+2s-2n}/bc;q)_{2r}}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\sum_{0\leq l}\frac{(-b/c,q^{2r-\lambda},-q^{1+s-n+r}/bc,-q^{s-n+r};q)_l}{(-bq^{n+\frac 12+r},-q^{\frac 12-\lambda-n+r}/c,-q^{1+2s-2n+2r}/bc,q;q)_l}q^l\\ &=\sum_{0\leq r,l}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,q^{2s-2n};q^2)_r(cq^{n+\frac 12};q)_r}{(q^{2s+2},q^2;q^2)_r(-bq^{n+\frac 12};q)_r(-q^{1+2s-2n}/bc;q)_{2r}}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\frac{(-b/c,-q^{1+s-n+r}/bc,-q^{s-n+r};q)_l}{(-bq^{n+\frac 12+r},-q^{1+2s-2n+2r}/bc,q;q)_l}q^l\\ &\qquad\cdot\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-bq^{n+\frac 12},-cq^{n+\frac 12};q)_{\lambda}}{(q,bq^{s+\frac 12},cq^{s+\frac 12},-bcq^{2n+1};q)_{\lambda}}q^{\lambda}\frac{(q^{-\lambda};q)_{2r}}{(-q^{\frac 12-\lambda-n}/b,-q^{\frac 12-\lambda-n}/c;q)_r}\frac{(q^{2r-\lambda};q)_l}{(-q^{\frac 12-\lambda-n+r}/c;q)_l}\\ &=\sum_{0\leq r,l}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,q^{2s-2n};q^2)_r(cq^{n+\frac 12};q)_r}{(q^{2s+2},q^2;q^2)_r(-bq^{n+\frac 12};q)_r(-q^{1+2s-2n}/bc;q)_{2r}}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\frac{(-b/c,-q^{1+s-n+r}/bc,-q^{s-n+r};q)_l}{(-bq^{n+\frac 12+r},-q^{1+2s-2n+2r}/bc,q;q)_l}q^l\gamma_{r,l,j} \end{align}
となる. ここで,
\begin{align} \gamma_{r,l,j}&:=\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-bq^{n+\frac 12},-cq^{n+\frac 12};q)_{\lambda}}{(q,bq^{s+\frac 12},cq^{s+\frac 12},-bcq^{2n+1};q)_{\lambda}}q^{\lambda}\frac{(q^{-\lambda};q)_{2r+l}}{(-q^{\frac 12-\lambda-n}/b;q)_r(-q^{\frac 12-\lambda-n}/c;q)_{r+l}}\\ &=(-1)^lb^rc^{r+l}q^{-\left(\frac 12-n\right)(2r+l)+r(r+l)}\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j};q)_{\lambda}(-bq^{n+\frac 12};q)_{\lambda-r}(-cq^{n+\frac 12};q)_{\lambda-r-l}}{(bq^{s+\frac 12},cq^{s+\frac 12},-bcq^{2n+1};q)_{\lambda}(q;q)_{\lambda-2r-l}}q^{\lambda}\\ &=(-1)^lb^rc^{r+l}q^{\left(\frac 12+n\right)(2r+l)+r(r+l)}\frac{(q^{-j},-bcq^{2s+j};q)_{2r+l}(-bq^{n+\frac 12};q)_{r+l}(-cq^{n+\frac 12};q)_{r}}{(bq^{s+\frac 12},cq^{s+\frac 12},-bcq^{2n+1};q)_{2r+l}}\\ &\qquad\cdot\Q43{q^{2r+l-j},-bcq^{2s+j+2r+l},-bq^{n+\frac 12+r+l},-cq^{n+\frac 12+r}}{bq^{s+\frac 12+2r+l},cq^{s+\frac 12+2r+l},-bcq^{2n+1+2r+l}}q \end{align}
である. ここで, Searsの変換公式 より
\begin{align} &\Q43{q^{2r+l-j},-bcq^{2s+j+2r+l},-bq^{n+\frac 12+r+l},-cq^{n+\frac 12+r}}{bq^{s+\frac 12+2r+l},cq^{s+\frac 12+2r+l},-bcq^{2n+1+2r+l}}q\\ &=\frac{(-cq^{s+\frac 12+2r+l},q^{2s-2n+2r+l};q)_{j-2r-l}}{(bq^{s+\frac 12+2r+l},-bcq^{2n+1+2r+l};q)_{j-2r-l}}(bq^{2n-s+\frac 12})^{j-2r-l}\Q43{q^{2r+l-j},-bcq^{2s+j+2r+l},-q^{s-n+r+l},-cq^{s-n+r}/b}{cq^{s+\frac 12+2r+l},-cq^{s+\frac 12+2r+l},q^{2s-2n+2r+l}}q \end{align}
であるから, これを代入して
\begin{align} \gamma_{r,l,j}&=(-1)^lb^rc^{r+l}q^{\left(\frac 12+n\right)(2r+l)+r(r+l)}\frac{(q^{-j},-bcq^{2s+j};q)_{2r+l}(-bq^{n+\frac 12};q)_{r+l}(-cq^{n+\frac 12};q)_{r}}{(bq^{s+\frac 12},cq^{s+\frac 12},-bcq^{2n+1};q)_{2r+l}}\\ &\qquad\cdot\frac{(-cq^{s+\frac 12+2r+l},q^{2s-2n+2r+l};q)_{j-2r-l}}{(bq^{s+\frac 12+2r+l},-bcq^{2n+1+2r+l};q)_{j-2r-l}}(bq^{2n-s+\frac 12})^{j-2r-l}\Q43{q^{2r+l-j},-bcq^{2s+j+2r+l},-q^{s-n+r+l},-cq^{s-n+r}/b}{cq^{s+\frac 12+2r+l},-cq^{s+\frac 12+2r+l},q^{2s-2n+2r+l}}q\\ &=(-1)^lb^{-r-l}c^{r+l}q^{\left(s-n\right)(2r+l)+r(r+l)}\frac{(q^{-j},-bcq^{2s+j};q)_{2r+l}(-bq^{n+\frac 12};q)_{r+l}(-cq^{n+\frac 12};q)_{r}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n};q)_{2r+l}}\\ &\qquad\cdot\frac{(-cq^{s+\frac 12},q^{2s-2n};q)_j}{(bq^{s+\frac 12},-bcq^{2n+1};q)_j}(bq^{2n-s+\frac 12})^{j}\sum_{0\leq \lambda}\frac{(q^{2r+l-j},-bcq^{2s+j+2r+l},-q^{s-n+r+l},-cq^{s-n+r}/b;q)_{\lambda}}{(cq^{s+\frac 12+2r+l},-cq^{s+\frac 12+2r+l},q^{2s-2n+2r+l},q;q)_{\lambda}}q^{\lambda}\\ &=(-1)^lb^{-r-l}c^{r+l}q^{\left(s-n\right)(2r+l)+r(r+l)}\frac{(-bq^{n+\frac 12};q)_{r+l}(-cq^{n+\frac 12};q)_{r}}{(-q^{s-n};q)_{r+l}(-cq^{s-n}/b;q)_r}\\ &\qquad\cdot\frac{(-cq^{s+\frac 12},q^{2s-2n};q)_j}{(bq^{s+\frac 12},-bcq^{2n+1};q)_j}(bq^{2n-s+\frac 12})^{j}\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j};q)_{\lambda}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n};q)_{\lambda}}q^{\lambda}\frac{(-q^{s-n};q)_{\lambda-r}(-cq^{s-n}/b;q)_{\lambda-r-l}}{(q;q)_{\lambda-2r-l}}\\ &=(-1)^lb^{-r-l}c^{r+l}q^{\left(s-n-1\right)(2r+l)+r(r+l)}\frac{(-bq^{n+\frac 12};q)_{r+l}(-cq^{n+\frac 12};q)_{r}}{(-q^{s-n};q)_{r+l}(-cq^{s-n}/b;q)_r}\\ &\qquad\cdot\frac{(-cq^{s+\frac 12},q^{2s-2n};q)_j}{(bq^{s+\frac 12},-bcq^{2n+1};q)_j}(bq^{2n-s+\frac 12})^{j}\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j};q)_{\lambda}(-q^{s-n};q)_{\lambda-r}(-cq^{s-n}/b;q)_{\lambda-r-l}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n};q)_{\lambda}(q;q)_{\lambda-2r-l}}q^{\lambda}\\ &=\frac{(-bq^{n+\frac 12};q)_{r+l}(-cq^{n+\frac 12};q)_{r}}{(-q^{s-n};q)_{r+l}(-cq^{s-n}/b;q)_r}\frac{(-cq^{s+\frac 12},q^{2s-2n};q)_j}{(bq^{s+\frac 12},-bcq^{2n+1};q)_j}(bq^{2n-s+\frac 12})^{j}\\ &\qquad\cdot\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-q^{s-n},-cq^{s-n}/b;q)_{\lambda}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n},q;q)_{\lambda}(q;q)_{\lambda-2r-l}}q^{\lambda}\frac{(q^{-\lambda};q)_{2r+l}}{(q^{1+n-s-\lambda};q)_r(-bq^{1+n-s-\lambda}/c;q)_{r+l}} \end{align}
を得る. よって,
\begin{align} \delta_j:=\frac{(-cq^{s+\frac 12},q^{2s-2n};q)_j}{(bq^{s+\frac 12},-bcq^{2n+1};q)_j}(bq^{2n-s+\frac 12})^{j} \end{align}
として,
\begin{align} F_j&=\delta_j\sum_{0\leq r,l}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,q^{2s-2n};q^2)_r(cq^{n+\frac 12};q)_r}{(q^{2s+2},q^2;q^2)_r(-bq^{n+\frac 12};q)_r(-q^{1+2s-2n}/bc;q)_{2r}}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\frac{(-b/c,-q^{1+s-n+r}/bc,-q^{s-n+r};q)_l}{(-bq^{n+\frac 12+r},-q^{1+2s-2n+2r}/bc,q;q)_l}q^l\frac{(-bq^{n+\frac 12};q)_{r+l}(-cq^{n+\frac 12};q)_{r}}{(-q^{s-n};q)_{r+l}(-cq^{s-n}/b;q)_r}\\ &\qquad\cdot\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-q^{s-n},-cq^{s-n}/b;q)_{\lambda}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n},q;q)_{\lambda}}q^{\lambda}\frac{(q^{-\lambda};q)_{2r+l}}{(q^{1+n-s-\lambda};q)_r(-bq^{1+n-s-\lambda}/c;q)_{r+l}}\\ &=\delta_j\sum_{0\leq r}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,q^{2s-2n};q^2)_r(cq^{n+\frac 12},-cq^{n+\frac 12};q)_r}{(q^{2s+2},q^2;q^2)_r(-q^{1+2s-2n}/bc;q)_{2r}(-q^{s-n},-cq^{s-n}/b;q)_r}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-q^{s-n},-cq^{s-n}/b;q)_{\lambda}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n},q;q)_{\lambda}(q^{1+n-s-\lambda};q)_r}q^{\lambda}\sum_{0\leq l}\frac{(-b/c,-q^{1+s-n+r}/bc;q)_l(q^{-\lambda};q)_{2r+l}}{(-q^{1+2s-2n+2r}/bc,q;q)_l(-bq^{1+n-s-\lambda}/c;q)_{r+l}}q^l\\ &=\delta_j\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-q^{s-n},-cq^{s-n}/b;q)_{\lambda}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n},q;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\sum_{0\leq r}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,c^2q^{2n+1};q^2)_r(q^{s-n};q)_r(q^{-\lambda};q)_{2r}}{(q^{2s+2},q^2;q^2)_r(-q^{1+2s-2n}/bc;q)_{2r}(-cq^{s-n}/b,q^{1+n-s-\lambda},-bq^{1+n-s-\lambda}/c;q)_r}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\sum_{0\leq l}\frac{(-b/c,-q^{1+s-n+r}/bc,q^{2r-\lambda};q)_{l}}{(-q^{1+2s-2n+2r}/bc,-bq^{1+n-s-\lambda+r}/c,q;q)_{l}}q^l \end{align}
ここで, $q$-Saalschützの和公式 より
\begin{align} \sum_{0\leq l}\frac{(-b/c,-q^{1+s-n+r}/bc,q^{2r-\lambda};q)_{l}}{(-q^{1+2s-2n+2r}/bc,-bq^{1+n-s-\lambda+r}/c,q;q)_{l}}q^l&=\frac{(q^{1+2s-2n+2r}/b^2,q^{s-n+r};q)_{\lambda-2r}}{(-q^{1+2s-2n+2r}/bc,-cq^{s-n+r}/b;q)_{\lambda-2r}} \end{align}
であるから, これを代入して,

\begin{align} F_j&=\delta_j\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-q^{s-n},-cq^{s-n}/b;q)_{\lambda}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n},q;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\sum_{0\leq r}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,c^2q^{2n+1};q^2)_r(q^{s-n};q)_r(q^{-\lambda};q)_{2r}}{(q^{2s+2},q^2;q^2)_r(-q^{1+2s-2n}/bc;q)_{2r}(-cq^{s-n}/b,q^{1+n-s-\lambda},-bq^{1+n-s-\lambda}/c;q)_r}\left(-\frac{bq^{2}}c\right)^r\\ &\qquad\cdot\frac{(q^{1+2s-2n+2r}/b^2,q^{s-n+r};q)_{\lambda-2r}}{(-q^{1+2s-2n+2r}/bc,-cq^{s-n+r}/b;q)_{\lambda-2r}}\\ &=\delta_j\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},-q^{s-n},-cq^{s-n}/b,q^{1+2s-2n}/b^2;q)_{\lambda}}{(cq^{s+\frac 12},-cq^{s+\frac 12},q^{2s-2n},-q^{1+2s-2n}/bc,q;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\sum_{0\leq r}\frac{(q^{1+2s-2n}/b^2,q^{2+2s-2n}/b^2c^2,c^2q^{2n+1};q^2)_r(q^{s-n};q)_{\lambda-r}(q^{-\lambda};q)_{2r}}{(q^{2s+2},q^2;q^2)_r(-cq^{s-n}/b;q)_{\lambda-r}(q^{1+n-s-\lambda},-bq^{1+n-s-\lambda}/c;q)_r(q^{1+2s-2n}/b^2;q)_{2r}}\left(-\frac{bq^{2}}c\right)^r\\ &=\delta_j\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},q^{1+2s-2n}/b^2;q)_{\lambda}(q^{2s-2n};q^2)_{\lambda}}{(q^{2s-2n},-q^{1+2s-2n}/bc,q;q)_{\lambda}(c^2q^{2s+1};q^2)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\Q43{q^{2+2s-2n}/b^2c^2,c^2q^{2n+1},q^{-\lambda},q^{1-\lambda}}{q^{2s+2},q^{2+2s-2n}/b^2,q^{2+2n-2s-2\lambda}}{q^2;q^{2}} \end{align}
ここで, Watsonの変換公式 より
\begin{align} &\Q43{q^{2+2s-2n}/b^2c^2,c^2q^{2n+1},q^{-\lambda},q^{1-\lambda}}{q^{2s+2},q^{2+2s-2n}/b^2,q^{2+2n-2s-2\lambda}}{q^2;q^{2}}\\ &=\frac{(q^{1+2s-2n},c^2q^{2s+2+\lambda},c^2q^{2s+1+\lambda},q^{2\lambda+2s-2n};q^2)_{\infty}}{(c^2q^{2s+2},q^{1+2s-2n+\lambda},q^{2s-2n+\lambda},c^2q^{2s+1+2\lambda};q^2)_{\infty}}\Q87{c^2q^{2s},cq^{s+2},-cq^{s+2},c^2,b^2c^2q^{2n},c^2q^{2n+1},q^{-\lambda},q^{1-\lambda}}{cq^s,-cq^s,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n},c^2q^{2s+2+\lambda},c^2q^{2s+1+\lambda}}{q^2;\frac{q^{4s-4n+2\lambda+2}}{b^2c^2}}\\ &=\frac{(c^2q^{2s+1};q^2)_{\lambda}(q^{2s-2n};q)_{\lambda}}{(c^2q^{2s+1};q)_{\lambda}(q^{2s-2n};q^2)_{\lambda}}\Q87{c^2q^{2s},cq^{s+2},-cq^{s+2},c^2,b^2c^2q^{2n},c^2q^{2n+1},q^{-\lambda},q^{1-\lambda}}{cq^s,-cq^s,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n},c^2q^{2s+2+\lambda},c^2q^{2s+1+\lambda}}{q^2;\frac{q^{4s-4n+2\lambda+2}}{b^2c^2}} \end{align}
であるから,
\begin{align} F_j&=\delta_j\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},q^{1+2s-2n}/b^2;q)_{\lambda}}{(-q^{1+2s-2n}/bc,c^2q^{2s+1},q;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\sum_{0\leq k}\frac{(1-c^2q^{4k+2s})(c^2q^{2s},c^2,b^2c^2q^{2n},c^2q^{2n+1},q^{-\lambda},q^{1-\lambda};q^2)_k}{(1-c^2q^{2s})(q^2,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n},c^2q^{2s+2+\lambda},c^2q^{2s+1+\lambda};q^2)_k}\left(\frac{q^{4s-4n+2\lambda+2}}{b^2c^2}\right)^k\\ &=\delta_j\sum_{0\leq k}\frac{(1-c^2q^{4k+2s})(c^2q^{2s},c^2,b^2c^2q^{2n},c^2q^{2n+1};q^2)_k}{(1-c^2q^{2s})(q^2,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n};q^2)_k}\left(\frac{q^{4s-4n+2}}{b^2c^2}\right)^k\\ &\qquad\cdot\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},q^{1+2s-2n}/b^2;q)_{\lambda}}{(-q^{1+2s-2n}/bc,c^2q^{2s+1},q;q)_{\lambda}}q^{\lambda}\frac{(q^{-\lambda};q)_{2k}}{(c^2q^{2s+1+\lambda};q)_{2k}}q^{2\lambda k}\\ &=\delta_j\sum_{0\leq k}\frac{(1-c^2q^{4k+2s})(c^2q^{2s},c^2,b^2c^2q^{2n},c^2q^{2n+1};q^2)_k}{(1-c^2q^{2s})(q^2,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n};q^2)_k}\left(\frac{q^{4s-4n+2}}{b^2c^2}\right)^kq^{\binom{2k}2}\\ &\qquad\cdot\sum_{0\leq \lambda}\frac{(q^{-j},-bcq^{2s+j},q^{1+2s-2n}/b^2;q)_{\lambda}}{(-q^{1+2s-2n}/bc;q)_{\lambda}(c^2q^{2s+1};q)_{\lambda+2k}(q;q)_{\lambda-2k}}q^{\lambda}\\ &=\delta_j\sum_{0\leq k}\frac{(1-c^2q^{4k+2s})(c^2q^{2s},c^2,b^2c^2q^{2n},c^2q^{2n+1};q^2)_k(q^{-j},-bcq^{2s+j},q^{1+2s-2n}/b^2;q)_{2k}}{(1-c^2q^{2s})(q^2,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n};q^2)_k(-q^{1+2s-2n}/bc;q)_{2k}(c^2q^{2s+1};q)_{4k}}\left(\frac{q^{4s-4n+4}}{b^2c^2}\right)^kq^{\binom{2k}2}\\ &\qquad\cdot\Q32{q^{2k-j},-bcq^{2s+j+2k},q^{1+2s-2n+2k}/b^2}{-q^{1+2s-2n+2k}/bc,c^2q^{2s+1+4k}}{q} \end{align}
となる. ここで, $q$-Saalschützの和公式 より
\begin{align} &\Q32{q^{2k-j},-bcq^{2s+j+2k},q^{1+2s-2n+2k}/b^2}{-q^{1+2s-2n+2k}/bc,c^2q^{2s+1+4k}}{q}\\ &=\frac{(b^2c^2q^{2n+2k},-b/c;q)_{j-2k}}{(c^2q^{2s+1+4k},-q^{1+2s-2n+2k}/bc;q)_{j-2k}}\left(\frac{q^{1+2s-2n+2k}}{b^2}\right)^{j-2k} \end{align}
であるから, これを代入して,
\begin{align} F_j&=\delta_j\frac{(b^2c^2q^{2n};q)_j}{(-q^{1+2s-2n}/bc;q)_{j}}\\ &\qquad\cdot\sum_{0\leq k}\frac{(1-c^2q^{4k+2s})(c^2q^{2s},c^2,b^2c^2q^{2n},c^2q^{2n+1};q^2)_k(q^{-j},-bcq^{2s+j},q^{1+2s-2n}/b^2;q)_{2k}(-b/c;q)_{j-2k}}{(1-c^2q^{2s})(q^2,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n};q^2)_k(c^2q^{2s+1};q)_{j+2k}(b^2c^2q^{2n};q)_{2k}}\left(\frac{q^{4s-4n+4}}{b^2c^2}\right)^kq^{\binom{2k}2}\left(\frac{q^{1+2s-2n+2k}}{b^2}\right)^{j-2k}\\ &=\delta_j\frac{(b^2c^2q^{2n},-b/c;q)_j}{(-q^{1+2s-2n}/bc,c^2q^{2s+1};q)_{j}}\left(\frac{q^{1+2s-2n}}{b^2}\right)^{j}\\ &\qquad\cdot\sum_{0\leq k}\frac{(1-c^2q^{4k+2s})(c^2q^{2s},c^2,b^2c^2q^{2n},c^2q^{2n+1};q^2)_k(q^{-j},-bcq^{2s+j},q^{1+2s-2n}/b^2;q)_{2k}}{(1-c^2q^{2s})(q^2,q^{2s+2},q^{2+2s-2n}/b^2,q^{1+2s-2n};q^2)_k(c^2q^{2s+1+j},b^2c^2q^{2n},-cq^{1-j}/b;q)_{2k}}q^{2k}\\ &=\delta_j\frac{(b^2c^2q^{2n},-b/c;q)_j}{(-q^{1+2s-2n}/bc,c^2q^{2s+1};q)_{j}}\left(\frac{q^{1+2s-2n}}{b^2}\right)^{j}\\ &\qquad\cdot\sum_{0\leq k}\frac{(1-c^2q^{4k+2s})(c^2q^{2s},c^2,c^2q^{2n+1},-bcq^{2s+j},-bcq^{2s+j+1},q^{1+2s-2n}/b^2,q^{-j},q^{1-j};q^2)_k}{(1-c^2q^{2s})(q^2,q^{2s+2},q^{1+2s-2n},-cq^{2-j}/b,-cq^{1-j}/b,b^2c^2q^{2n+1},c^2q^{2s+2+j},c^2q^{2s+1+j};q^2)_k}q^{2k}\\ &=\frac{(-cq^{s+\frac 12},q^{2s-2n},b^2c^2q^{2n},-b/c;q)_j}{(bq^{s+\frac 12},-bcq^{2n+1},-q^{1+2s-2n}/bc,c^2q^{2s+1};q)_j}\left(\frac{q^{\frac 32+s}}{b}\right)^{j}\\ &\qquad\cdot\Q{10}9{c^2q^{2s},cq^{s+1},-cq^{s+1},c^2,c^2q^{2n+1},-bcq^{2s+j},-bcq^{2s+j+1},q^{1+2s-2n}/b^2,q^{-j},q^{1-j}}{cq^s,-cq^s,q^{2s+2},q^{1+2s-2n},-cq^{2-j}/b,-cq^{1-j}/b,b^2c^2q^{2n+1},c^2q^{2s+2+j},c^2q^{2s+1+j}}{q^2;q^2} \end{align}
を得る. つまり, 以下が得られたことになる.
\begin{align} F_j&=\frac{(-cq^{s+\frac 12},q^{2s-2n},b^2c^2q^{2n},-b/c;q)_j}{(bq^{s+\frac 12},-bcq^{2n+1},-q^{1+2s-2n}/bc,c^2q^{2s+1};q)_j}\left(\frac{q^{\frac 32+s}}{b}\right)^{j}\\ &\qquad\cdot\Q{10}9{c^2q^{2s},cq^{s+1},-cq^{s+1},c^2,c^2q^{2n+1},-bcq^{2s+j},-bcq^{2s+j+1},q^{1+2s-2n}/b^2,q^{-j},q^{1-j}}{cq^s,-cq^s,q^{2s+2},q^{1+2s-2n},-cq^{2-j}/b,-cq^{1-j}/b,b^2c^2q^{2n+1},c^2q^{2s+2+j},c^2q^{2s+1+j}}{q^2;q^2} \end{align}
これによって, 連続$q$-Jacobi多項式の線形化公式の係数が${}_{10}\phi_9$で表されることが分かった. ここまでで主要な計算は終わったわけであるが, 続きの考察は 次の記事 で行いたいと思う.