連続q-Jacobi多項式の線形化公式1

$x:=\cos\theta$として,
\begin{align} r_n(x)=r_n(x;a,b,c,d)&:=\Q43{q^{-n},abcdq^{n-1},ae^{i\theta},ae^{-i\theta}}{ab,ac,ad}q\\ w(x)=w(x;a,b,c,d)&:=\frac 1{\sqrt{1-x^2}}\frac{(e^{2i\theta},e^{-2i\theta};q)_{\infty}}{(ae^{i\theta},ae^{-i\theta},be^{i\theta},be^{-i\theta},ce^{i\theta},ce^{-i\theta},de^{i\theta},de^{-i\theta};q)_{\infty}} \end{align}
と定義する. 連続$q$-Jacobi多項式は本質的に
\begin{align} r_n(x;q^{\frac 12},b,c,-q^{\frac 12}) \end{align}
の場合である. 今回は連続$q$-Jacobi多項式の線形化公式を示したRahmanの1981年の論文について解説していきたいと思う.
\begin{align} g_k=g_k(m,n;a,b,c,d):=\int_{-1}^1r_m(x)r_n(x)r_k(x)w(x)\,dx \end{align}
とする. このとき, Askey-Wilson多項式の対称性 から$r_n(x;a,b,c,d)$は$b,c,d$に関して対称であり,
\begin{align} r_n(x;a,b,c,d)=\frac{(bc,bd;q)_n}{(ac,ad;q)_n}\left(\frac ab\right)^nr_n(x;b,a,c,d) \end{align}
が成り立つ. これより,
\begin{align} f_k=f_k(m,n;a,b,c,d):=\int_{-1}^1r_m(x;b,a,c,d)r_n(x;c,b,a,d)r_k(x;a,b,c,d)w(x)\,dx \end{align}
としたとき,
\begin{align} g_k=\frac{(bc,bd;q)_m(cb,cd;q)_n}{(ac,ad;q)_m(ab,ad;q)_n}\left(\frac ab\right)^m\left(\frac ac\right)^nf_k \end{align}
が成り立つことが分かる. $f_k$は$r$の定義より,
\begin{align} f_k&=\int_{-1}^1r_m(x;b,a,c,d)r_n(x;c,b,a,d)r_k(x;a,b,c,d)w(x)\,dx\\ &=\sum_{0\leq \lambda,\mu,\nu}\frac{(q^{-k},abcdq^{k-1};q)_{\lambda}}{(q,ab,ac,ad;q)_{\lambda}}\frac{(q^{-m},abcdq^{m-1};q)_{\mu}}{(q,ba,bc,bd;q)_{\mu}}\frac{(q^{-n},abcdq^{n-1};q)_{\nu}}{(q,ca,cb,cd;q)_{\nu}}q^{\lambda+\mu+\nu}I(\lambda,\mu,\nu) \end{align}
と表される. ここで,
\begin{align} I(\lambda,\mu,\nu)&=\int_{-1}^1\frac{dx}{\sqrt{1-x^2}}\frac{(e^{2i\theta},e^{-2i\theta};q)_{\infty}}{(ae^{i\theta}q^{\lambda},ae^{-i\theta}q^{\lambda},be^{i\theta}q^{\mu},be^{-i\theta}q^{\mu},ce^{i\theta}q^{\nu},ce^{-i\theta}q^{\nu},de^{i\theta},de^{-i\theta};q)_{\infty}} \end{align}
である. Askey-Wilson積分 より, これは
\begin{align} I(\lambda,\mu,\nu)&=\frac{2\pi(abcdq^{\lambda+\mu+\nu};q)_{\infty}}{(abq^{\lambda+\mu},acq^{\lambda+\nu},adq^{\lambda},bcq^{\mu+\nu},bdq^{\mu},cdq^{\nu};q)_{\infty}}\\ &=h_0\frac{(ab;q)_{\lambda+\mu}(ac;q)_{\lambda+\nu}(bc;q)_{\mu+\nu}(ad;q)_{\lambda}(bd;q)_{\mu}(cd;q)_{\nu}}{(abcd;q)_{\lambda+\mu+\nu}}\\ h_0&:=\frac{2\pi(abcd;q)_{\infty}}{(q,ab,ac,ad,bc,bd,cd;q)_{\infty}} \end{align}
となる. これを代入すると,
\begin{align} f_k&=h_0\sum_{0\leq \lambda,\mu,\nu}\frac{(q^{-k},abcdq^{k-1};q)_{\lambda}}{(q,ab,ac,ad;q)_{\lambda}}\frac{(q^{-m},abcdq^{m-1};q)_{\mu}}{(q,ba,bc,bd;q)_{\mu}}\frac{(q^{-n},abcdq^{n-1};q)_{\nu}}{(q,ca,cb,cd;q)_{\nu}}q^{\lambda+\mu+\nu}\\ &\qquad\cdot\frac{(ab;q)_{\lambda+\mu}(ac;q)_{\lambda+\nu}(bc;q)_{\mu+\nu}(ad;q)_{\lambda}(bd;q)_{\mu}(cd;q)_{\nu}}{(abcd;q)_{\lambda+\mu+\nu}}\\ &=h_0\sum_{0\leq \lambda,\mu}\frac{(q^{-k},abcdq^{k-1};q)_{\lambda}}{(q,ab;q)_{\lambda}}\frac{(q^{-m},abcdq^{m-1};q)_{\mu}}{(q,ba;q)_{\mu}}\frac{(ab;q)_{\lambda+\mu}}{(abcd;q)_{\lambda+\mu}}q^{\lambda+\mu}\Q43{q^{-n},abcdq^{n-1},acq^{\lambda},bcq^{\mu}}{ac,bc,abcdq^{\lambda+\mu}}q \end{align}
を得る. ここで, Searsの変換公式 より,
\begin{align} \Q43{q^{-n},abcdq^{n-1},acq^{\lambda},bcq^{\mu}}{ac,bc,abcdq^{\lambda+\mu}}q&=\frac{(ad,q^{-\lambda-\mu};q)_n}{(bc,abcdq^{\lambda+\mu};q)_n}(bcq^{\lambda+\mu})^n\Q43{q^{-n},abcdq^{n-1},q^{-\lambda},aq^{-\mu}/b}{ac,ad,q^{-\lambda-\mu}}q \end{align}
となる. この右辺は$(q^{-\mu-\mu};q)_n$が掛かっているので, $\lambda+\mu< n$のとき, $0$になっている. よって, $m=n-s, k=s+j$として, $0\leq s\leq n, 0\leq j\leq 2n-2s$とすると, $0$にならない$\lambda$の範囲は$0\leq \mu\leq n-s, \lambda+\mu\geq n$から$s\leq \lambda\leq s+j$となる.
\begin{align} g_{s+j}&=g_{s+j}(n-s,n;a,b,c,d)\\ &=h_0\frac{(bc,bd;q)_{n-s}(cb,cd;q)_n}{(ac,ad;q)_{n-s}(ab,ad;q)_n}\left(\frac ab\right)^{n-s}\left(\frac ac\right)^n\\ &\qquad\cdot\sum_{\lambda=s}^{s+j}\sum_{\mu=n-\lambda}^{n-s}\frac{(q^{-s-j},abcdq^{s+j-1};q)_{\lambda}}{(q,ab;q)_{\lambda}}\frac{(q^{s-n},abcdq^{n-s-1};q)_{\mu}}{(q,ba;q)_{\mu}}\frac{(ab;q)_{\lambda+\mu}}{(abcd;q)_{\lambda+\mu}}q^{\lambda+\mu}\\ &\qquad\cdot\frac{(ad,q^{-\lambda-\mu};q)_n}{(bc,abcdq^{\lambda+\mu};q)_n}(bcq^{\lambda+\mu})^n\Q43{q^{-n},abcdq^{n-1},q^{-\lambda},aq^{-\mu}/b}{ac,ad,q^{-\lambda-\mu}}q\\ &=h_0\frac{(bc,bd;q)_{n-s}(cd;q)_n}{(ac,ad;q)_{n-s}(ab;q)_n}a^{2n-s}b^s\\ &\qquad\cdot\sum_{\lambda=0}^{j}\sum_{\mu=n-\lambda-s}^{n-s}\frac{(q^{-s-j},abcdq^{s+j-1};q)_{\lambda+s}}{(q,ab;q)_{\lambda+s}}\frac{(q^{s-n},abcdq^{n-s-1};q)_{\mu}}{(q,ba;q)_{\mu}}\frac{(ab;q)_{\lambda+\mu+s}}{(abcd;q)_{\lambda+\mu+s}}q^{(n+1)(\lambda+\mu+s)}\\ &\qquad\cdot\frac{(q^{-\lambda-\mu-s};q)_n}{(abcdq^{\lambda+\mu+s};q)_n}\Q43{q^{-n},abcdq^{n-1},q^{-\lambda-s},aq^{-\mu}/b}{ac,ad,q^{-\lambda-\mu-s}}q\qquad\lambda\mapsto\lambda+s\\ &=h_0\frac{(bc,bd;q)_{n-s}(cd;q)_n}{(ac,ad;q)_{n-s}(ab;q)_n}a^{2n-s}b^s\\ &\qquad\cdot\sum_{\lambda=0}^{j}\sum_{\mu=0}^{\lambda}\frac{(q^{-s-j},abcdq^{s+j-1};q)_{\lambda+s}}{(q,ab;q)_{\lambda+s}}\frac{(q^{s-n},abcdq^{n-s-1};q)_{n-s-\lambda+\mu}}{(q,ab;q)_{n-s-\lambda+\mu}}\frac{(ab;q)_{n+\mu}(q^{-n-\mu};q)_n}{(abcd;q)_{2n+\mu}}q^{(n+1)(n+\mu)}\\ &\qquad\cdot\Q43{q^{-n},abcdq^{n-1},q^{-\lambda-s},aq^{\lambda+s-n-\mu}/b}{ac,ad,q^{-n-\mu}}q\qquad\mu\mapsto n-s-\lambda+\mu\\ &=h_0\frac{(bc,bd;q)_{n-s}(cd;q)_n}{(ac,ad;q)_{n-s}(abcd;q)_{2n}}a^{2n-s}b^s\frac{(q^{-s-j},abcdq^{s+j-1};q)_{s}}{(q,ab;q)_{s}}\frac{(q^{s-n},abcdq^{n-s-1};q)_{n-s}}{(q,ab;q)_{n-s}}\\ &\qquad\cdot\sum_{\lambda=0}^{j}\sum_{\mu=0}^{\lambda}\frac{(q^{-j},abcdq^{2s+j-1};q)_{\lambda}}{(q^{s+1},abq^s;q)_{\lambda}}\frac{(1,abcdq^{2n-2s-1};q)_{-\lambda+\mu}}{(q^{1+n-s},abq^{n-s};q)_{-\lambda+\mu}}\frac{(abq^n;q)_{\mu}(q;q)_{n+\mu}}{(abcdq^{2n},q;q)_{\mu}}(-1)^nq^{\binom{n+1}2+\mu}\\ &\qquad\cdot\Q43{q^{-n},abcdq^{n-1},q^{-\lambda-s},aq^{\lambda+s-n-\mu}/b}{ac,ad,q^{-n-\mu}}q\\ &=h_0\frac{(-1)^nq^{\binom{n+1}2}(bc,bd,q^{s-n},abcdq^{n-s-1};q)_{n-s}(q^{-s-j},abcdq^{s+j-1};q)_{s}(q,cd;q)_n}{(q,ab,ac,ad;q)_{n-s}(q,ab;q)_{s}(abcd;q)_{2n}}a^{2n-s}b^se_j \end{align}
となる. ここで,
\begin{align} e_j&:=\sum_{\lambda=0}^{j}\frac{(q^{-j},abcdq^{2s+j-1},q^{s-n},q^{1+s-n}/ab;q)_{\lambda}}{(q^{s+1},abq^s,q,q^{2+2s-2n}/abcd;q)_{\lambda}}\left(\frac{q^2}{cd}\right)^{\lambda}\\ &\qquad\cdot\sum_{\mu=0}^{\lambda}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n,q^{n+1};q)_{\mu}}{(q^{1+n-s-\lambda},abq^{n-s-\lambda},abcdq^{2n},q;q)_{\mu}}q^{\mu}\\ &\qquad\cdot\Q43{q^{-n},abcdq^{n-1},q^{-\lambda-s},aq^{\lambda+s-n-\mu}/b}{ac,ad,q^{-n-\mu}}q \end{align}
である. ここで, Searsの変換公式 より,
\begin{align} &\Q43{q^{-n},abcdq^{n-1},q^{-\lambda-s},aq^{\lambda+s-n-\mu}/b}{ac,ad,q^{-n-\mu}}q\\ &=\frac{(bcq^{n-\lambda-s},bdq^{n-\lambda-s};q)_{\lambda+s}}{(ac,ad;q)_{\lambda+s}}\left(\frac{aq^{\lambda+s-n}}b\right)^{\lambda+s}\Q43{abcdq^{n-1},q^{-\lambda-s},q^{-\mu},bq^{-\lambda-s}/a}{q^{-n-\mu},bcq^{n-\lambda-s},bdq^{n-\lambda-s}}q \end{align}
となる. ここで, Watsonの変換公式 より,
\begin{align} &\Q43{abcdq^{n-1},q^{-\lambda-s},q^{-\mu},bq^{-\lambda-s}/a}{q^{-n-\mu},bcq^{n-\lambda-s},bdq^{n-\lambda-s}}q\\ &=\frac{(q^{1+n-\lambda-s},abcdq^{2n};q)_{\mu}}{(abcdq^{2n-\lambda-s},q^{n+1};q)_{\mu}}\Q87{abcdq^{2n-\lambda-s-1},q\sqrt{abcdq^{2n-\lambda-s-1}},-q\sqrt{abcdq^{2n-\lambda-s-1}},acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s},q^{-\mu}}{\sqrt{abcdq^{2n-\lambda-s-1}},-\sqrt{abcdq^{2n-\lambda-s-1}},bcq^{n-\lambda-s},bdq^{n-\lambda-s},q^{1+n-\lambda-s},abcdq^{2n},abcdq^{2n-\lambda+\mu-s}}{\frac{bq^{1+n-s-\lambda+\mu}}{a}}\\ &=\frac{(q^{1+n-\lambda-s},abcdq^{2n};q)_{\mu}}{(abcdq^{2n-\lambda-s},q^{n+1};q)_{\mu}}\sum_{0\leq i}\frac{(1-abcdq^{2i+2n-\lambda-s-1})(abcdq^{2n-\lambda-s-1},acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s},q^{-\mu};q)_i}{(1-abcdq^{2n-\lambda-s-1})(q,bcq^{n-\lambda-s},bdq^{n-\lambda-s},q^{1+n-\lambda-s},abcdq^{2n},abcdq^{2n-\lambda+\mu-s};q)_i}\left(\frac{bq^{1+n-s-\lambda+\mu}}{a}\right)^i \end{align}
であるから,
\begin{align} &\sum_{\mu=0}^{\lambda}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n,q^{n+1};q)_{\mu}}{(q^{1+n-s-\lambda},abq^{n-s-\lambda},abcdq^{2n},q;q)_{\mu}}q^{\mu}\\ &\qquad\cdot\Q43{q^{-n},abcdq^{n-1},q^{-\lambda-s},aq^{\lambda+s-n-\mu}/b}{ac,ad,q^{-n-\mu}}q\\ &=\frac{(bcq^{n-\lambda-s},bdq^{n-\lambda-s};q)_{\lambda+s}}{(ac,ad;q)_{\lambda+s}}\left(\frac{aq^{\lambda+s-n}}b\right)^{\lambda+s}\sum_{\mu=0}^{\lambda}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n,q^{n+1};q)_{\mu}}{(q^{1+n-s-\lambda},abq^{n-s-\lambda},abcdq^{2n},q;q)_{\mu}}q^{\mu}\\ &\qquad\cdot\frac{(q^{1+n-\lambda-s},abcdq^{2n};q)_{\mu}}{(abcdq^{2n-\lambda-s},q^{n+1};q)_{\mu}}\sum_{0\leq i}\frac{(1-abcdq^{2i+2n-\lambda-s-1})(abcdq^{2n-\lambda-s-1},acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s},q^{-\mu};q)_i}{(1-abcdq^{2n-\lambda-s-1})(q,bcq^{n-\lambda-s},bdq^{n-\lambda-s},q^{1+n-\lambda-s},abcdq^{2n},abcdq^{2n-\lambda+\mu-s};q)_i}\left(\frac{bq^{1+n-s-\lambda+\mu}}{a}\right)^i\\ &=\frac{(bcq^{n-\lambda-s},bdq^{n-\lambda-s};q)_{\lambda+s}}{(ac,ad;q)_{\lambda+s}}\left(\frac{aq^{\lambda+s-n}}b\right)^{\lambda+s}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2i+2n-\lambda-s-1})(abcdq^{2n-\lambda-s-1},acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s};q)_i}{(1-abcdq^{2n-\lambda-s-1})(q,bcq^{n-\lambda-s},bdq^{n-\lambda-s},q^{1+n-\lambda-s},abcdq^{2n};q)_i}\left(-\frac{bq^{1+n-s-\lambda}}{a}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq \mu}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{\mu}}{(abq^{n-s-\lambda};q)_{\mu}(abcdq^{2n-\lambda-s};q)_{\mu+i}(q;q)_{\mu-i}}q^{\mu} \end{align}
ここで, $q$-Saalschützの和公式 より
\begin{align} &\sum_{0\leq \mu}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{\mu}}{(abq^{n-s-\lambda};q)_{\mu}(abcdq^{2n-\lambda-s};q)_{\mu+i}(q;q)_{\mu-i}}q^{\mu}\\ &=\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{i}}{(abq^{n-s-\lambda};q)_i(abcdq^{2n-\lambda-s};q)_{2i}}q^i\Q32{q^{i-\lambda},abcdq^{2n-2s-1-\lambda+i},abq^{n+i}}{abq^{n-s-\lambda+i},abcdq^{2n-\lambda-s+2i}}q\\ &=\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{i}}{(abq^{n-s-\lambda};q)_i(abcdq^{2n-\lambda-s};q)_{2i}}q^i\frac{(q^{-s-\lambda},cdq^{n-\lambda-s+i};q)_{\lambda-i}}{(abq^{n-s-\lambda+i},abcdq^{2n-\lambda-s+2i};q)_{\lambda-i}}(abq^{n+i})^{\lambda-i}\\ &=\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{i}}{(abq^{n-s-\lambda};q)_i(abcdq^{2n-\lambda-s};q)_{\lambda+i}}q^i\frac{(q^{-s-\lambda},q^{1+s-n}/cd;q)_{\lambda-i}}{(q^{1+s-n}/ab;q)_{\lambda-i}}(cdq^{n+i})^{\lambda-i}\\ &=\frac{(q^{-s-\lambda},q^{1+s-n}/cd;q)_{\lambda}}{(abcdq^{2n-\lambda-s},q^{1+s-n}/ab;q)_{\lambda}}(cdq^n)^{\lambda}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{i}}{(q^{s+1},cdq^{n-s-\lambda},abcdq^{2n-s};q)_{i}}\left(-\frac {q^{\lambda-n+s+1}}{ab}\right)^iq^{-\binom{i}2}\\ &=\frac{(q^{s+1},q^{1+s-n}/cd;q)_{\lambda}}{(q^{1+s-2n}/abcd,q^{1+s-n}/ab;q)_{\lambda}}(abq^n)^{-\lambda}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{i}}{(q^{s+1},cdq^{n-s-\lambda},abcdq^{2n-s};q)_{i}}\left(-\frac {q^{\lambda-n+s+1}}{ab}\right)^iq^{-\binom{i}2} \end{align}
であるから, これを代入すると,
\begin{align} &\sum_{\mu=0}^{\lambda}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n,q^{n+1};q)_{\mu}}{(q^{1+n-s-\lambda},abq^{n-s-\lambda},abcdq^{2n},q;q)_{\mu}}q^{\mu}\\ &\qquad\cdot\Q43{q^{-n},abcdq^{n-1},q^{-\lambda-s},aq^{\lambda+s-n-\mu}/b}{ac,ad,q^{-n-\mu}}q\\ &=\frac{(bcq^{n-\lambda-s},bdq^{n-\lambda-s};q)_{\lambda+s}}{(ac,ad;q)_{\lambda+s}}\left(\frac{aq^{\lambda+s-n}}b\right)^{\lambda+s}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2i+2n-\lambda-s-1})(abcdq^{2n-\lambda-s-1},acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s};q)_i}{(1-abcdq^{2n-\lambda-s-1})(q,bcq^{n-\lambda-s},bdq^{n-\lambda-s},q^{1+n-\lambda-s},abcdq^{2n};q)_i}\left(-\frac{bq^{1+n-s-\lambda}}{a}\right)^iq^{\binom i2}\\ &\qquad\cdot\frac{(q^{s+1},q^{1+s-n}/cd;q)_{\lambda}}{(q^{1+s-2n}/abcd,q^{1+s-n}/ab;q)_{\lambda}}(abq^n)^{-\lambda}\frac{(q^{-\lambda},abcdq^{2n-2s-1-\lambda},abq^n;q)_{i}}{(q^{s+1},cdq^{n-s-\lambda},abcdq^{2n-s};q)_{i}}\left(-\frac {q^{\lambda-n+s+1}}{ab}\right)^iq^{-\binom{i}2}\\ &=\frac{(q^{1-n}/bc,q^{1-n}/bd;q)_{\lambda+s}}{(ac,ad;q)_{\lambda+s}}\left(abcdq^{n-1}\right)^{\lambda+s}\frac{(q^{s+1},q^{1+s-n}/cd;q)_{\lambda}}{(q^{1+s-2n}/abcd,q^{1+s-n}/ab;q)_{\lambda}}(abq^n)^{-\lambda}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2i+2n-\lambda-s-1})(abcdq^{2n-\lambda-s-1},abq^n,acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s},abcdq^{2n-2s-1-\lambda},q^{-\lambda};q)_i}{(1-abcdq^{2n-\lambda-s-1})(q,bcq^{n-\lambda-s},bdq^{n-\lambda-s},cdq^{n-s-\lambda},q^{1+n-\lambda-s},abcdq^{2n},q^{s+1},abcdq^{2n-s};q)_i}\left(\frac{q^2}{a^2}\right)^i\\ &=\frac{(q^{1-n}/bc,q^{1-n}/bd;q)_{s}}{(ac,ad;q)_{s}}\left(abcdq^{n-1}\right)^{s}\frac{(q^{s+1},q^{1+s-n}/bc,q^{1+s-n}/bd,q^{1+s-n}/cd;q)_{\lambda}}{(acq^s,adq^s,q^{1+s-2n}/abcd,q^{1+s-n}/ab;q)_{\lambda}}\left(\frac{cd}q\right)^{\lambda}\\ &\qquad\cdot\Q{10}9{abcdq^{2n-\lambda-s-1},q\sqrt{abcdq^{2n-\lambda-s-1}},-q\sqrt{abcdq^{2n-\lambda-s-1}},abq^n,acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s},abcdq^{2n-2s-1-\lambda},q^{-\lambda}}{\sqrt{abcdq^{2n-\lambda-s-1}},-\sqrt{abcdq^{2n-\lambda-s-1}},bcq^{n-\lambda-s},bdq^{n-\lambda-s},cdq^{n-s-\lambda},q^{1+n-\lambda-s},abcdq^{2n},q^{s+1},abcdq^{2n-s}}{\frac{q^2}{a^2}} \end{align}
を得る. よって,
\begin{align} e_j&=\frac{(q^{1-n}/bc,q^{1-n}/bd;q)_{s}}{(ac,ad;q)_{s}}\left(abcdq^{n-1}\right)^{s}\sum_{\lambda=0}^{j}\frac{(q^{-j},abcdq^{2s+j-1},q^{s-n},q^{1+s-n}/bc,q^{1+s-n}/bd,q^{1+s-n}/cd;q)_{\lambda}}{(q,abq^s,acq^s,adq^s,q^{2+2s-2n}/abcd,q^{1+s-2n}/abcd;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\Q{10}9{abcdq^{2n-\lambda-s-1},q\sqrt{abcdq^{2n-\lambda-s-1}},-q\sqrt{abcdq^{2n-\lambda-s-1}},abq^n,acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s},abcdq^{2n-2s-1-\lambda},q^{-\lambda}}{\sqrt{abcdq^{2n-\lambda-s-1}},-\sqrt{abcdq^{2n-\lambda-s-1}},bcq^{n-\lambda-s},bdq^{n-\lambda-s},cdq^{n-s-\lambda},q^{1+n-\lambda-s},abcdq^{2n},q^{s+1},abcdq^{2n-s}}{\frac{q^2}{a^2}} \end{align}
となる. これより,
\begin{align} F_j&:=\sum_{\lambda=0}^{j}\frac{(q^{-j},abcdq^{2s+j-1},q^{s-n},q^{1+s-n}/bc,q^{1+s-n}/bd,q^{1+s-n}/cd;q)_{\lambda}}{(q,abq^s,acq^s,adq^s,q^{2+2s-2n}/abcd,q^{1+s-2n}/abcd;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\Q{10}9{abcdq^{2n-\lambda-s-1},q\sqrt{abcdq^{2n-\lambda-s-1}},-q\sqrt{abcdq^{2n-\lambda-s-1}},abq^n,acq^n,adq^n,abcdq^{n-1},q^{-\lambda-s},abcdq^{2n-2s-1-\lambda},q^{-\lambda}}{\sqrt{abcdq^{2n-\lambda-s-1}},-\sqrt{abcdq^{2n-\lambda-s-1}},bcq^{n-\lambda-s},bdq^{n-\lambda-s},cdq^{n-s-\lambda},q^{1+n-\lambda-s},abcdq^{2n},q^{s+1},abcdq^{2n-s}}{\frac{q^2}{a^2}} \end{align}
とすると,
\begin{align} g_{s+j}&=h_0\frac{(-1)^nq^{\binom{n+1}2}(bc,bd,q^{s-n},abcdq^{n-s-1};q)_{n-s}(q^{-s-j},abcdq^{s+j-1};q)_{s}(q,cd;q)_n}{(q,ab,ac,ad;q)_{n-s}(q,ab;q)_{s}(abcd;q)_{2n}}a^{2n-s}b^s\\ &\qquad\cdot\frac{(q^{1-n}/bc,q^{1-n}/bd;q)_{s}}{(ac,ad;q)_{s}}\left(abcdq^{n-1}\right)^{s}F_j\\ &=h_0\frac{(-1)^nq^{\binom{n+1}2}(q^{s-n},abcdq^{n-s-1};q)_{n-s}(q^{-s-j},abcdq^{s+j-1};q)_{s}(q,bc,bd,cd;q)_n}{(q,ab,ac,ad;q)_{n-s}(q,ab,ac,ad;q)_{s}(abcd;q)_{2n}}a^{2n}q^{s(s-n)}F_j\\ &=h_0\frac{q^{-js}(abcdq^{n-s-1};q)_{n-s}(q^{j+1},abcdq^{s+j-1};q)_{s}(q,bc,bd,cd;q)_n}{(ab,ac,ad;q)_{n-s}(q,ab,ac,ad;q)_{s}(abcd;q)_{2n}}a^{2n}F_j\\ \end{align}
ここまでは$a,b,c,d$が一般の場合で成り立つ公式である. ここで, $a=q^{\frac 12},d=-q^{\frac 12}$とすると,
\begin{align} F_j&=\sum_{\lambda=0}^{j}\frac{(q^{-j},-bcq^{2s+j},q^{s-n},q^{1+s-n}/bc,-q^{\frac 12+s-n}/b,-q^{\frac 12+s-n}/c;q)_{\lambda}}{(q,bq^{s+\frac 12},cq^{s+\frac 12},-q^{s+1},-q^{1+2s-2n}/bc,-q^{s-2n}/bc;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\Q{10}9{-bcq^{2n-\lambda-s},q\sqrt{-bcq^{2n-\lambda-s}},-q\sqrt{-bcq^{2n-\lambda-s}},bq^{n+\frac 12},cq^{n+\frac 12},-q^{n+1},-bcq^{n},q^{-\lambda-s},-bcq^{2n-2s-\lambda},q^{-\lambda}}{\sqrt{-bcq^{2n-\lambda-s}},-\sqrt{-bcq^{2n-\lambda-s}},bcq^{n-\lambda-s},-bq^{\frac 12+n-\lambda-s},-cq^{\frac 12+n-s-\lambda},q^{1+n-\lambda-s},-bcq^{2n+1},q^{s+1},-bcq^{2n-s+1}}{q} \end{align}
となる. Baileyの変換公式( 前の記事 の定理1)より
\begin{align} &\Q{10}9{-bcq^{2n-\lambda-s},q\sqrt{-bcq^{2n-\lambda-s}},-q\sqrt{-bcq^{2n-\lambda-s}},bq^{n+\frac 12},cq^{n+\frac 12},-q^{n+1},-bcq^{n},q^{-\lambda-s},-bcq^{2n-2s-\lambda},q^{-\lambda}}{\sqrt{-bcq^{2n-\lambda-s}},-\sqrt{-bcq^{2n-\lambda-s}},bcq^{n-\lambda-s},-bq^{\frac 12+n-\lambda-s},-cq^{\frac 12+n-s-\lambda},q^{1+n-\lambda-s},-bcq^{2n+1},q^{s+1},-bcq^{2n-s+1}}{q}\\ &=\frac{(-bcq^{2n-\lambda-s+1},-q^{\frac 12-\lambda-n}/b,-q^{\frac 12-\lambda-n}/c,q^{s+1};q)_{\lambda}}{(-q^{1-\lambda},-bq^{\frac 12+n-\lambda-s},-cq^{\frac 12+n-\lambda-s},-bcq^{2n+1};q)_{\lambda}}\left(bcq^{2n-s}\right)^{\lambda}\\ &\qquad\cdot\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\\ &=\frac{(-q^{s-2n}/bc,-bq^{n+\frac 12},-cq^{n+\frac 12},-q^{s+1};q)_{\lambda}}{(-1,-q^{\frac 12+s-n}/b,-q^{\frac 12+s-n}/c,-bcq^{2n+1};q)_{\lambda}}\\ &\qquad\cdot\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}
であるから,
\begin{align} F_j&=\sum_{\lambda=0}^{j}\frac{(q^{-j},-bcq^{2s+j},q^{s-n},q^{1+s-n}/bc,-q^{\frac 12+s-n}/b,-q^{\frac 12+s-n}/c;q)_{\lambda}}{(q,bq^{s+\frac 12},cq^{s+\frac 12},-q^{s+1},-q^{1+2s-2n}/bc,-q^{s-2n}/bc;q)_{\lambda}}q^{\lambda}\\ &\qquad\cdot\frac{(-q^{s-2n}/bc,-bq^{n+\frac 12},-cq^{n+\frac 12},-q^{s+1};q)_{\lambda}}{(-1,-q^{\frac 12+s-n}/b,-q^{\frac 12+s-n}/c,-bcq^{2n+1};q)_{\lambda}}\\ &\qquad\cdot\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_{\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}\\ &\qquad\cdot\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}
を得る. この後
\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 \end{align}
をさらに変形していくわけであるが, 記事が長くなってきたので続きの計算は 次の記事 で解説したいと思う.