連続q-Jacobi多項式の線形化公式2: K_λ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}
今回はこの$K_{\lambda}$を変形していきたいと思う.
\begin{align} 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\\ &=\sum_{0\leq k}\frac{(1+q^{2k-\lambda})(-q^{-\lambda},q^{-\lambda},-q^{-\lambda-s},q^{-\lambda-s},bq^{n+\frac 12},-bq^{n+\frac 12};q)_k}{(1+q^{-\lambda})(q,-q,q^{s+1},-q^{s+1},-q^{\frac 12-\lambda-n}/b,q^{\frac 12-\lambda-n}/b;q)_k}\\ &\qquad\cdot\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12},-q^{1+s-n}/bc,-q^{s-n};q)_k}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c,bcq^{n-\lambda-s},q^{1+n-s-\lambda};q)_k}q^k\\ &=\sum_{0\leq k}\frac{(1+q^{2k-\lambda})(q^{-2\lambda},q^{-2\lambda-2s},b^2q^{2n+1};q^2)_k}{(1+q^{-\lambda})(q^2,q^{2s+2},q^{1-2\lambda-2n}/b^2;q^2)_k}\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12},-q^{1+s-n}/bc,-q^{s-n};q)_k}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c,bcq^{n-\lambda-s},q^{1+n-s-\lambda};q)_k}q^k \end{align}
ここで, $q$-Saalschützの和公式 より
\begin{align} \frac{(q^{-2\lambda-2s},b^2q^{2n+1};q^2)_k}{(q^{2s+2},q^{1-2\lambda-2n}/b^2;q^2)_k}&=(b^2q^{2n-2s-1})^k\sum_{r=0}^k\frac{(q^{-2k},q^{2k-2\lambda},q^{1+2s-2n}/b^2;q^2)_r}{(q^{2s+2},q^{1-2\lambda-2n}/b^2,q^2;q^2)_r}q^{2r} \end{align}
であるから, これを代入すると,
\begin{align} K_{\lambda}&=\sum_{0\leq k}\frac{(1+q^{2k-\lambda})(q^{-2\lambda};q^2)_k}{(1+q^{-\lambda})(q^2;q^2)_k}\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12},-q^{1+s-n}/bc,-q^{s-n};q)_k}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c,bcq^{n-\lambda-s},q^{1+n-s-\lambda};q)_k}q^k\\ &\qquad\cdot(b^2q^{2n-2s-1})^k\sum_{r=0}^k\frac{(q^{-2k},q^{2k-2\lambda},q^{1+2s-2n}/b^2;q^2)_r}{(q^{2s+2},q^{1-2\lambda-2n}/b^2,q^2;q^2)_r}q^{2r}\\ &=\sum_{0\leq r}\frac{(q^{1+2s-2n}/b^2;q^2)_r}{(q^{2s+2},q^{1-2\lambda-2n}/b^2,q^2;q^2)_r}(-1)^rq^{2\binom{r+1}2}\\ &\qquad\cdot\sum_{0\leq k}\frac{(1+q^{2k-\lambda})(q^{-2\lambda};q^2)_{k+r}}{(1+q^{-\lambda})(q^2;q^2)_{k-r}}\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12},-q^{1+s-n}/bc,-q^{s-n};q)_k}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c,bcq^{n-\lambda-s},q^{1+n-s-\lambda};q)_k}(b^2q^{2n-2s-2r})^k\\ &=\sum_{0\leq r}\frac{1+q^{2r-\lambda}}{1+q^{-\lambda}}\frac{(q^{1+2s-2n}/b^2;q^2)_r(q^{-2\lambda};q^2)_{2r}}{(q^{2s+2},q^{1-2\lambda-2n}/b^2,q^2;q^2)_r}\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12},-q^{1+s-n}/bc,-q^{s-n};q)_r}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c,bcq^{n-\lambda-s},q^{1+n-s-\lambda};q)_r}(-b^2q^{2n-2s-2r})^rq^{2\binom{r+1}2}\\ &\qquad\cdot\sum_{0\leq k}\frac{(1+q^{2k+2r-\lambda})(q^{4r-2\lambda};q^2)_{k}}{(1+q^{2r-\lambda})(q^2;q^2)_{k}}\frac{(q^{\frac 12-\lambda-n+r}/b,cq^{n+\frac 12+r},-q^{1+s-n+r}/bc,-q^{s-n+r};q)_k}{(-bq^{n+\frac 12+r},-q^{\frac 12-\lambda-n+r}/c,bcq^{n-\lambda-s+r},q^{1+n-s-\lambda+r};q)_k}(b^2q^{2n-2s-2r})^k\\ &=\sum_{0\leq r}\frac{1+q^{2r-\lambda}}{1+q^{-\lambda}}\frac{(q^{1+2s-2n}/b^2;q^2)_r(q^{-2\lambda};q^2)_{2r}}{(q^{2s+2},q^{1-2\lambda-2n}/b^2,q^2;q^2)_r}\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12},-q^{1+s-n}/bc,-q^{s-n};q)_r}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c,bcq^{n-\lambda-s},q^{1+n-s-\lambda};q)_r}(-b^2q^{1+2n-2s-r})^r\\ &\qquad\cdot\Q87{-q^{2r-\lambda},q\sqrt{-q^{2r-\lambda}},-q\sqrt{-q^{2r-\lambda}},q^{2r-\lambda},q^{\frac 12-\lambda-n+r}/b,cq^{n+\frac 12+r},-q^{1+s-n+r}/bc,-q^{s-n+r}}{\sqrt{-q^{2r-\lambda}},-\sqrt{-q^{2r-\lambda}},-q,-bq^{n+\frac 12+r},-q^{\frac 12-\lambda-n+r}/c,bcq^{n-\lambda-s+r},q^{1+n-s-\lambda+r}}{b^2q^{2n-2s-2r}} \end{align}
ここで, Watsonの変換公式 より
\begin{align} &\Q87{-q^{2r-\lambda},q\sqrt{-q^{2r-\lambda}},-q\sqrt{-q^{2r-\lambda}},q^{2r-\lambda},q^{\frac 12-\lambda-n+r}/b,cq^{n+\frac 12+r},-q^{1+s-n+r}/bc,-q^{s-n+r}}{\sqrt{-q^{2r-\lambda}},-\sqrt{-q^{2r-\lambda}},-q,-bq^{n+\frac 12+r},-q^{\frac 12-\lambda-n+r}/c,bcq^{n-\lambda-s+r},q^{1+n-s-\lambda+r}}{b^2q^{2n-2s-2r}}\\ &=\frac{(-q^{1+2r-\lambda},-bcq^{2n-2s-\lambda};q)_{\lambda-2r}}{(bcq^{n-\lambda-s+r},q^{1+n-s-\lambda+r};q)_{\lambda-2r}}\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}\\ &=\frac{(-1,-q^{1+2s-2n+2r}/bc;q)_{\lambda-2r}}{( q^{1-n+s+r}/bc,q^{s-n+r};q)_{\lambda-2r}}\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}
ここで,
\begin{align} &\frac{(-1,-q^{1+2s-2n+2r}/bc;q)_{\lambda-2r}}{( q^{1-n+s+r}/bc,q^{s-n+r};q)_{\lambda-2r}}\\ &=\frac{(-1;q)_{\lambda-2r}(-q^{1+2s-2n}/bc;q)_{\lambda}( q^{1-n+s}/bc,q^{s-n};q)_{r}}{(-q^{1+2s-2n}/bc;q)_{2r}( q^{1-n+s}/bc,q^{s-n};q)_{\lambda-r}}\\ &=\frac{(-1,-q^{1+2s-2n}/bc;q)_{\lambda}}{( q^{1-n+s}/bc,q^{s-n};q)_{\lambda}}\frac{( q^{1-n+s}/bc,q^{s-n},bcq^{n-s-\lambda},q^{1+n-s-\lambda};q)_{r}}{(-q^{1+2s-2n}/bc,-q^{1-\lambda};q)_{2r}}\left(\frac{q^{1+2s-2n}}{bc}\right)^rq^{r^2} \end{align}
であるから,
\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{1+q^{2r-\lambda}}{1+q^{-\lambda}}\frac{(q^{1+2s-2n}/b^2;q^2)_r(q^{-2\lambda};q^2)_{2r}}{(q^{2s+2},q^{1-2\lambda-2n}/b^2,q^2;q^2)_r}\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12},-q^{1+s-n}/bc,-q^{s-n};q)_r}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c,bcq^{n-\lambda-s},q^{1+n-s-\lambda};q)_r}(-b^2q^{1+2n-2s-r})^r\\ &\qquad\cdot\frac{( q^{1-n+s}/bc,q^{s-n},bcq^{n-s-\lambda},q^{1+n-s-\lambda};q)_{r}}{(-q^{1+2s-2n}/bc,-q^{1-\lambda};q)_{2r}}\left(\frac{q^{1+2s-2n}}{bc}\right)^rq^{r^2}\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}\\ &=\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^2)_r(q^{-2\lambda};q^2)_{2r}}{(q^{2s+2},q^{1-2\lambda-2n}/b^2,q^2;q^2)_r}\frac{(q^{\frac 12-\lambda-n}/b,cq^{n+\frac 12};q)_r}{(-bq^{n+\frac 12},-q^{\frac 12-\lambda-n}/c;q)_r(-q^{1+2s-2n}/bc,-q^{-\lambda};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}\\ &=\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}
つまり, 以下の表示が得られた.
\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}
この$K_{\lambda}$はRahmanの論文に書かれているものと微妙に違っているので, あっているのかどうかが怪しいのであるが, とりあえずRahmanの論文に書かれているものと全く同じように計算を進めることができたので, 最終的にはRahmanの結果に一致することを期待して計算を進めようと思う. 次の記事 で$F_j$の計算を行いたいと思う.