Askey-Wilson関数の漸化式

\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}
とする. 前の記事 で, Askey-Wilson多項式が満たす漸化式を示した. 今回はAskey-Wilson関数
\begin{align} R_{\alpha}(z)&=\frac{(abq^{\alpha},acq^{\alpha},adq^{\alpha},bcdq^{\alpha}/z;q)_{\infty}}{(bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},azq^{\alpha};q)_{\infty}}\left(\frac az\right)^{\alpha}\\ &\qquad\cdot W(bcd/zq;b/z,c/z,d/z,abcdq^{\alpha-1},q^{-\alpha};zq/a)\\ S_{\alpha}(z)&=\frac{(abcdq^{2\alpha},bzq^{\alpha+1},czq^{\alpha+1},dzq^{\alpha+1},bcdzq^{\alpha};q)_{\infty}}{(bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},q^{\alpha+1},bcdzq^{2\alpha+1};q)_{\infty}}(az)^{\alpha}\\ &\qquad\cdot W(bcdzq^{2\alpha};bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},q^{\alpha+1},zq/a;az)\\ \end{align}
が漸化式
\begin{align} 2xh_{\alpha}(z)&=A_{\alpha}h_{\alpha+1}(z)+B_{\alpha}h_{\alpha}(x)+C_{\alpha}h_{\alpha-1}(z) \end{align}
を満たすことを示す. ここで,
\begin{align} A_{\alpha}&:=\frac{a^{-1}(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})}\\ C_{\alpha}&:=\frac{a(1-bcq^{\alpha-1})(1-bdq^{\alpha-1})(1-cdq^{\alpha-1})(1-q^{\alpha})}{(1-abcdq^{2\alpha-2})(1-abcdq^{2\alpha-1})}\\ B_{\alpha}&:=a+a^{-1}-A_{\alpha}-C_{\alpha}\\ x&:=\frac{z+z^{-1}}2 \end{align}
である.

$R$の漸化式

まず, $R$が漸化式を満たすことを示す. 前の記事 の定理2において, $a,b,c,d,e,f$を$bcd/zq,q^{-\alpha},abcdq^{\alpha-1},b/z,c/z,d/z$とすると,
\begin{align} f_{\alpha}(z):=W(bcd/zq;b/z,c/z,d/z,abcdq^{\alpha-1},q^{-\alpha};zq/a) \end{align}
として,
\begin{align} 0&=\frac{(1-abcdq^{\alpha-1})(1-q^{-\alpha}/az)(1-q^{1-\alpha}/az)(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}(f_{\alpha+1}(z)-f_{\alpha}(z))\\ &-\frac{(1-q^{-\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-q^{1-\alpha}/ab)(1-q^{1-\alpha}/ac)(1-q^{1-\alpha}/ad)}{1-q^{2-2\alpha}/abcd}(f_{\alpha-1}(z)-f_{\alpha}(z))\\ &+\frac{z}{a}(q^{-\alpha}-abcdq^{\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)f_{\alpha}(z)\\ &=\frac{q^{1-2\alpha}}{a^2z^2}\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}(f_{\alpha+1}(z)-f_{\alpha}(z))\\ &+\frac{q^{1-2\alpha}}{a^2}\frac{(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}(f_{\alpha-1}(z)-f_{\alpha}(z))\\ &+\frac{zq^{-\alpha}}{a}(1-abcdq^{2\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)f_{\alpha}(z)\\ &=\frac{q^{1-2\alpha}}{a^2z^2}\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}f_{\alpha+1}(z)\\ &+\frac{q^{1-2\alpha}}{a^2}\frac{(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}f_{\alpha-1}(z)\\ &+\frac{zq^{-\alpha}}{a}(1-abcdq^{2\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)f_{\alpha}(z)\\ &-\frac{q^{1-2\alpha}}{a^2z^2}\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}f_{\alpha}(z)\\ &-\frac{q^{1-2\alpha}}{a^2}\frac{(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}f_{\alpha}(z)\\ &=\frac{q^{1-2\alpha}}{a^2z^2}\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}\\ &\qquad\cdot \frac{(bcq^{\alpha+1},bdq^{\alpha+1},cdq^{\alpha+1},azq^{\alpha+1};q)_{\infty}}{(abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1},bcdq^{\alpha+1}/z;q)_{\infty}}\left(\frac za\right)^{\alpha+1}R_{\alpha+1}(z)\\ &+\frac{q^{1-2\alpha}}{a^2}\frac{(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}\\ &\qquad\cdot \frac{(bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},azq^{\alpha-1};q)_{\infty}}{(abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1},bcdq^{\alpha-1}/z;q)_{\infty}}\left(\frac za\right)^{\alpha-1}R_{\alpha-1}(z)\\ &+\frac{(bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},azq^{\alpha};q)_{\infty}}{(abq^{\alpha},acq^{\alpha},adq^{\alpha},bcdq^{\alpha}/z;q)_{\infty}}\left(\frac za\right)^{\alpha}\\ &\qquad\cdot\bigg(\frac{zq^{-\alpha}}{a}(1-abcdq^{2\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)\\ &-\frac{q^{1-2\alpha}}{a^2z^2}\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}\\ &-\frac{q^{1-2\alpha}}{a^2}\frac{(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}\bigg)R_{\alpha}(z)\\ &=\frac{(bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},azq^{\alpha};q)_{\infty}}{(abq^{\alpha},acq^{\alpha},adq^{\alpha},bcdq^{\alpha}/z;q)_{\infty}}\left(\frac za\right)^{\alpha}\\ &\cdot\Bigg(\frac{q^{1-2\alpha}}{a^3z}\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-bcdq^{\alpha}/z)(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})}{1-abcdq^{2\alpha}}R_{\alpha+1}(z)\\ &+\frac{q^{1-2\alpha}}{az}\frac{(1-q^{\alpha})(1-bcdq^{\alpha}/z)(1-azq^{\alpha-1})(1-bcq^{\alpha-1})(1-bdq^{\alpha-1})(1-cdq^{\alpha-1})}{1-abcdq^{2\alpha-2}}R_{\alpha-1}(z)\\ &+\bigg(\frac{zq^{-\alpha}}{a}(1-abcdq^{2\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)\\ &-\frac{q^{1-2\alpha}}{a^2z^2}\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}\\ &-\frac{q^{1-2\alpha}}{a^2}\frac{(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}\bigg)R_{\alpha}(z)\Bigg) \end{align}
となるので,
\begin{align} 0&=\frac{a^{-1}(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-bcdq^{\alpha}/z)(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})}{1-abcdq^{2\alpha}}R_{\alpha+1}(z)\\ &+\frac{a(1-q^{\alpha})(1-bcdq^{\alpha}/z)(1-azq^{\alpha-1})(1-bcq^{\alpha-1})(1-bdq^{\alpha-1})(1-cdq^{\alpha-1})}{1-abcdq^{2\alpha-2}}R_{\alpha-1}(z)\\ &+\bigg(az^2q^{\alpha-1}(1-abcdq^{2\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)\\ &-\frac{z^{-1}(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}\\ &-\frac{z(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}\bigg)R_{\alpha}(z) \end{align}
となる. ここで, $R_{\alpha}(z)$の係数は
\begin{align} &az^2q^{\alpha-1}(1-abcdq^{2\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)\\ &-\frac{z^{-1}(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})}{1-abcdq^{2\alpha}}\\ &-\frac{z(1-q^{\alpha})(1-bcdq^{\alpha-1}/z)(1-bcdq^{\alpha}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})}{1-abcdq^{2\alpha-2}}\\ &=az^2q^{\alpha-1}(1-abcdq^{2\alpha-1})(1-q/az)(1-b/z)(1-c/z)(1-d/z)\\ &-\frac{(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(z^{-1}(1-azq^{\alpha})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})-a^{-1}(1-bcdq^{\alpha}/z)(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha}))}{1-abcdq^{2\alpha}}\\ &-\frac{(1-q^{\alpha})(1-bcdq^{\alpha}/z)(z(1-bcdq^{\alpha-1}/z)(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})-a(1-azq^{\alpha-1})(1-bcq^{\alpha-1})(1-bdq^{\alpha-1})(1-cdq^{\alpha-1}))}{1-abcdq^{2\alpha-2}}\\ &-\frac{a^{-1}(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-bcdq^{\alpha}/z)(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})}{1-abcdq^{2\alpha}}\\ &-\frac{a(1-q^{\alpha})(1-bcdq^{\alpha}/z)(1-azq^{\alpha-1})(1-bcq^{\alpha-1})(1-bdq^{\alpha-1})(1-cdq^{\alpha-1})}{1-abcdq^{2\alpha-2}}\\ &=(1-abcdq^{2\alpha-1})(1-azq^{\alpha-1})(1-bcdq^{\alpha}/z)(a+a-z-z^{-1})\\ &-\frac{a^{-1}(1-abcdq^{\alpha-1})(1-azq^{\alpha-1})(1-bcdq^{\alpha}/z)(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})}{1-abcdq^{2\alpha}}\\ &-\frac{a(1-q^{\alpha})(1-bcdq^{\alpha}/z)(1-azq^{\alpha-1})(1-bcq^{\alpha-1})(1-bdq^{\alpha-1})(1-cdq^{\alpha-1})}{1-abcdq^{2\alpha-2}} \end{align}
となる(最後の等号は数式処理システムを用いた). これを代入して両辺を$(1-abcdq^{2\alpha-1})(1-azq^{\alpha-1})(1-bcdq^{\alpha}/z)$で割ると
\begin{align} 0&=A_{\alpha}R_{\alpha+1}(z)+C_{\alpha}R_{\alpha-1}(z)+(a+a^{-1}-A_{\alpha}-C_{\alpha}-z-z^{-1})R_{\alpha}(z)\\ &=A_{\alpha}R_{\alpha+1}(z)+(B_{\alpha}-2x)R_{\alpha}(z)+C_{\alpha}R_{\alpha-1}(z) \end{align}
となる. つまり以下が得られた.

$S$の漸化式

Nassrallrah-Rahman積分( 前の記事 の定理2)は$z=e^{i\theta}$として,
\begin{align} \int_{0}^{\pi}v(z;a,b,c,d,e;A)\,d\theta&=\frac{2\pi(Aa,Ab,Ac,Ad,Ae,abcde/A;q)_{\infty}}{(q,ab,ac,ad,ae,bc,bd,be,cd,ce,de,A^2;q)_{\infty}}\\ &\qquad\cdot W(A^2/q;A/a,A/b,A/c,A/d,A/e;abcde/A)\\ v(z;a,b,c,d,e;A)&:=\left|\frac{(z^2,Az;q)_{\infty}}{(az,bz,cz,dz,ez;q)_{\infty}}\right|^2 \end{align}
と表される. ここで,
\begin{align} (1-Az)(1-Az^{-1})=(1-Aa)(1-A/a)+A(1-az)(1-az^{-1})/a \end{align}
であることを用いると, $(Az,A/z;q)_{\infty}=(1-Az)(1-Az^{-1})(Aqz,Aq/z;q)_{\infty}$より,
\begin{align} &\int_0^{\pi}v(z;a,b,c,d,e;A)\,d\theta\\ &=(1-Aa)(1-A/a)\int_0^{\pi}v(z;a,b,c,d,e;Aq)\,d\theta\\ &+\frac Aa\int_0^{\pi}v(z;aq,b,c,d,e;Aq)\,d\theta\\ \end{align}
となる. これとNassrallah-Rahman積分より
\begin{align} W(A;a)&=\frac{(1-A/a)(1-A^2)(1-A^2q)(1-abcde/Aq)}{(1-Ab)(1-Ac)(1-Ad)(1-Ae)}W(Aq;a)\\ &+\frac Aa\frac{(1-A^2)(1-A^2q)(1-ab)(1-ac)(1-ad)(1-ae)}{(1-Aa)(1-Aaq)(1-Ab)(1-Ac)(1-Ad)(1-Ae)}W(Aq;aq) \end{align}
を得る. ここで,
\begin{align} W(A;a):=W(A^2/q;A/a,A/b,A/c,A/d,A/e;abcde/A) \end{align}
である. 上の$A,a$を$A/q,a/q$に置き換えると
\begin{align} W(A/q;a/q)&=\frac{(1-A/a)(1-A^2/q^2)(1-A^2/q)(1-abcde/Aq)}{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}W(A;a/q)\\ &+\frac Aa\frac{(1-A^2/q^2)(1-A^2/q)(1-ab/q)(1-ac/q)(1-ad/q)(1-ae/q)}{(1-Aa/q^2)(1-Aa/q)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}W(A;a) \end{align}
を得る. ここで,
\begin{align} \frac{(Aq/a;q)_n}{(Aa/q;q)_n}-\frac{(A/a;q)_n}{(Aa;q)_n}q^n&=\frac{(Aq/a;q)_{n-1}((1-Aq^n/a)(1-Aaq^{n-1})-(1-A/a)(1-Aa/q)q^n)}{(Aa/q;q)_{n+1}}\\ &=\frac{(Aq/a;q)_{n-1}(1-q^n)(1-A^2q^{n-1})}{(Aa/q;q)_{n+1}} \end{align}
であるから,
\begin{align} &W(A;a/q)-W(A;a)\\ &=\sum_{0\leq n}\frac{(1-A^2q^{2n-1})(A^2,A/b,A/c,A/d,A/e;q)_n(Aq/a;q)_{n-1}}{(q;q)_{n-1}(Ab,Ac,Ad,Ae;q)_n(Aa/q;q)_{n+1}}\left(\frac{abcde}{Aq}\right)^n\\ &=\frac{(1-A^2)(1-A^2q)(1-A/b)(1-A/c)(1-A/d)(1-A/e)}{(1-Aa/q)(1-Aa)(1-Ab)(1-Ac)(1-Ad)(1-Ae)}\frac{abcde}{Aq}W(Aq;a) \end{align}
となる. これを用いて, 上の$W(A;a/q)$を消去すると,
\begin{align} W(A/q;a/q)&=\frac{(1-A/a)(1-A^2/q^2)(1-A^2/q)(1-abcde/Aq)}{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\\ &\qquad\cdot\left(W(A;a)+\frac{(1-A^2)(1-A^2q)(1-A/b)(1-A/c)(1-A/d)(1-A/e)}{(1-Aa/q)(1-Aa)(1-Ab)(1-Ac)(1-Ad)(1-Ae)}\frac{abcde}{Aq}W(Aq;a)\right)\\ &+\frac Aa\frac{(1-A^2/q^2)(1-A^2/q)(1-ab/q)(1-ac/q)(1-ad/q)(1-ae/q)}{(1-Aa/q^2)(1-Aa/q)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}W(A;a)\\ &=\frac{(A^2/q^2;q)_4(1-A/a)(1-A/b)(1-A/c)(1-A/d)(1-A/e)(1-abcde/Aq)}{(Aa/q,Ab/q,Ac/q,Ad/q,Ae/q;q)_2}\frac{abcde}{Aq}W(Aq;a)\\ &+\bigg(\frac{(1-A/a)(1-A^2/q^2)(1-A^2/q)(1-abcde/Aq)}{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\\ &\qquad+\frac Aa\frac{(1-A^2/q^2)(1-A^2/q)(1-ab/q)(1-ac/q)(1-ad/q)(1-ae/q)}{(1-Aa/q^2)(1-Aa/q)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\bigg)W(A;a) \end{align}
を得る. ここで,　先ほどの式
\begin{align} W(A;a)&=\frac{(1-A/a)(1-A^2)(1-A^2q)(1-abcde/Aq)}{(1-Ab)(1-Ac)(1-Ad)(1-Ae)}W(Aq;a)\\ &+\frac Aa\frac{(1-A^2)(1-A^2q)(1-ab)(1-ac)(1-ad)(1-ae)}{(1-Aa)(1-Aaq)(1-Ab)(1-Ac)(1-Ad)(1-Ae)}W(Aq;aq) \end{align}
を用いて上の$W(Aq;a)$を消去すると,
\begin{align} W(A/q;a/q) &=\frac{(1-A^2/q^2)(1-A^2/q)(1-A/b)(1-A/c)(1-A/d)(1-A/e)}{(1-Aa/q)(1-Aa)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\frac{abcde}{Aq}\\ &\qquad\left(W(A;a)-\frac Aa\frac{(1-A^2)(1-A^2q)(1-ab)(1-ac)(1-ad)(1-ae)}{(1-Aa)(1-Aaq)(1-Ab)(1-Ac)(1-Ad)(1-Ae)}W(Aq;aq)\right)\\ &+\bigg(\frac{(1-A/a)(1-A^2/q^2)(1-A^2/q)(1-abcde/Aq)}{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\\ &\qquad+\frac Aa\frac{(1-A^2/q^2)(1-A^2/q)(1-ab/q)(1-ac/q)(1-ad/q)(1-ae/q)}{(1-Aa/q^2)(1-Aa/q)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\bigg)W(A;a)\\ &=-\frac{(1-A^2/q^2)(1-A^2/q)(1-A/b)(1-A/c)(1-A/d)(1-A/e)}{(1-Aa/q)(1-Aa)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\frac{bcde}{q}\\ &\qquad\cdot\frac{(1-A^2)(1-A^2q)(1-ab)(1-ac)(1-ad)(1-ae)}{(1-Aa)(1-Aaq)(1-Ab)(1-Ac)(1-Ad)(1-Ae)}W(Aq;aq)\\ &+\bigg(\frac{(1-A/a)(1-A^2/q^2)(1-A^2/q)(1-abcde/Aq)}{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\\ &\qquad+\frac Aa\frac{(1-A^2/q^2)(1-A^2/q)(1-ab/q)(1-ac/q)(1-ad/q)(1-ae/q)}{(1-Aa/q^2)(1-Aa/q)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\\ &\qquad+\frac{(1-A^2/q^2)(1-A^2/q)(1-A/b)(1-A/c)(1-A/d)(1-A/e)}{(1-Aa/q)(1-Aa)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\frac{abcde}{Aq}\bigg)W(A;a)\\ &=-\frac{(A^2/q^2;q)_4(1-A/b)(1-A/c)(1-A/d)(1-A/e)(1-ab)(1-ac)(1-ad)(1-ae)}{(Aa/q,Aa,Ab/q,Ac/q,Ad/q,Ae/q;q)_2}\frac{bcde}{q}W(Aq;aq)\\ &+\bigg(\frac{(1-A/a)(1-A^2/q^2)(1-A^2/q)(1-abcde/Aq)}{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\\ &\qquad+\frac Aa\frac{(1-A^2/q^2)(1-A^2/q)(1-ab/q)(1-ac/q)(1-ad/q)(1-ae/q)}{(1-Aa/q^2)(1-Aa/q)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\\ &\qquad+\frac{(1-A^2/q^2)(1-A^2/q)(1-A/b)(1-A/c)(1-A/d)(1-A/e)}{(1-Aa/q)(1-Aa)(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}\frac{abcde}{Aq}\bigg)W(A;a) \end{align}
両辺に
\begin{align} \frac{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}{(1-A^2/q^2)(1-A^2/q)} \end{align}
を掛けて
\begin{align} &\frac{(1-Ab/q)(1-Ac/q)(1-Ad/q)(1-Ae/q)}{(1-A^2/q^2)(1-A^2/q)}W(A/q;a/q)\\ &+\frac{(A^2;q)_2(1-A/b)(1-A/c)(1-A/d)(1-A/e)(1-ab)(1-ac)(1-ad)(1-ae)}{(Aa/q,Aa;q)_2(1-Ab)(1-Ac)(1-Ad)(1-Ae)}\frac{bcde}{q}W(Aq;aq)\\ &=\bigg((1-A/a)(1-abcde/Aq)+\frac Aa\frac{(1-ab/q)(1-ac/q)(1-ad/q)(1-ae/q)}{(1-Aa/q^2)(1-Aa/q)}\\ &\qquad+\frac{(1-A/b)(1-A/c)(1-A/d)(1-A/e)}{(1-Aa/q)(1-Aa)}\frac{abcde}{Aq}\bigg)W(A;a) \end{align}
を得る.
\begin{align} g_{\alpha}(z):=W(bcdzq^{2\alpha};bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},q^{\alpha+1},zq/a;az) \end{align}
とする. 上の式で, $A,a,b,c,d,e$を$q^{\alpha}\sqrt{bcdzq}, aq^{\alpha}\sqrt{bcd/zq},\sqrt{dzq/bc},\sqrt{czq/bd},\sqrt{bzq/cd},\sqrt{bcdz/q}$とすると,
\begin{align} 0&=\frac{(1-bzq^{\alpha})(1-czq^{\alpha})(1-dzq^{\alpha})(1-bcdzq^{\alpha-1})}{(1-bcdzq^{2\alpha-1})(1-bcdzq^{2\alpha})}g_{\alpha-1}(z)\\ &+\frac{(1-bcdzq^{2\alpha+1})(1-bcdzq^{2\alpha+2})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})(1-q^{\alpha+1})(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})^2(1-abcdq^{2\alpha+1})(1-bzq^{\alpha+1})(1-czq^{\alpha+1})(1-dzq^{\alpha+1})(1-bcdzq^{\alpha})}z^2g_{\alpha+1}(z)\\ &-\bigg((1-zq/a)(1-az/q)+\frac{zq}a\frac{(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})(1-abcdq^{\alpha-2})}{(1-abcdq^{2\alpha-2})(1-abcdq^{2\alpha-1})}\\ &\qquad+\frac{az}{q}\frac{(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})(1-q^{\alpha+1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})}\bigg)g_{\alpha}(z)\\ &=\frac{(1-bzq^{\alpha})(1-czq^{\alpha})(1-dzq^{\alpha})(1-bcdzq^{\alpha-1})}{(1-bcdzq^{2\alpha-1})(1-bcdzq^{2\alpha})}\\ &\qquad\cdot\frac{(bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{\alpha},bcdzq^{2\alpha-1};q)_{\infty}}{(abcdq^{2\alpha-2},bzq^{\alpha},czq^{\alpha},dzq^{\alpha},bcdzq^{\alpha-1};q)_{\infty}}(az)^{-\alpha+1}S_{\alpha-1}(z)\\ &+\frac{(1-bcdzq^{2\alpha+1})(1-bcdzq^{2\alpha+2})(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})(1-q^{\alpha+1})(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})^2(1-abcdq^{2\alpha+1})(1-bzq^{\alpha+1})(1-czq^{\alpha+1})(1-dzq^{\alpha+1})(1-bcdzq^{\alpha})}z^2\\ &\qquad\cdot\frac{(bcq^{\alpha+1},bdq^{\alpha+1},cdq^{\alpha+1},q^{\alpha+2},bcdzq^{2\alpha+3};q)_{\infty}}{(abcdq^{2\alpha+2},bzq^{\alpha+2},czq^{\alpha+2},dzq^{\alpha+2},bcdzq^{\alpha+1};q)_{\infty}}(az)^{-\alpha-1}S_{\alpha+1}(z)\\ &-\bigg((1-zq/a)(1-az/q)+\frac{zq}a\frac{(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})(1-abcdq^{\alpha-2})}{(1-abcdq^{2\alpha-2})(1-abcdq^{2\alpha-1})}\\ &\qquad+\frac{az}{q}\frac{(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})(1-q^{\alpha+1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})}\bigg)\\ &\qquad\cdot\frac{(bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},q^{\alpha+1},bcdzq^{2\alpha+1};q)_{\infty}}{(abcdq^{2\alpha},bzq^{\alpha+1},czq^{\alpha+1},dzq^{\alpha+1},bcdzq^{\alpha};q)_{\infty}}(az)^{-\alpha}S_{\alpha}(z)\\ &=\frac{(bcq^{\alpha},bdq^{\alpha},cdq^{\alpha},q^{\alpha+1},bcdzq^{2\alpha+1};q)_{\infty}}{(abcdq^{2\alpha},bzq^{\alpha+1},czq^{\alpha+1},dzq^{\alpha+1},bcdzq^{\alpha};q)_{\infty}}(az)^{-\alpha}\Bigg(\frac{(1-bcq^{\alpha-1})(1-bdq^{\alpha-1})(1-cdq^{\alpha-1})(1-q^{\alpha})}{(1-abcdq^{2\alpha-2})(1-abcdq^{2\alpha-1})}azS_{\alpha-1}(z)\\ &+\frac{(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})}a^{-1}zS_{\alpha+1}(z)\\ &-\bigg((1-zq/a)(1-az/q)+\frac{zq}a\frac{(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})(1-abcdq^{\alpha-2})}{(1-abcdq^{2\alpha-2})(1-abcdq^{2\alpha-1})}\\ &\qquad+\frac{az}{q}\frac{(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})(1-q^{\alpha+1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})}\bigg)S_{\alpha}(z)\Bigg) \end{align}
よって,
\begin{align} 0&=A_{\alpha}S_{\alpha+1}(z)+C_{\alpha}S_{\alpha-1}(z)\\ &\qquad-\bigg(z^{-1}(1-zq/a)(1-az/q)+\frac{q}a\frac{(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})(1-abcdq^{\alpha-2})}{(1-abcdq^{2\alpha-2})(1-abcdq^{2\alpha-1})}\\ &\qquad+\frac{a}{q}\frac{(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})(1-q^{\alpha+1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})}\bigg)S_{\alpha}(z) \end{align}
となり, $S_{\alpha}(z)$の係数は
\begin{align} &z^{-1}(1-zq/a)(1-az/q)+\frac{q}a\frac{(1-abq^{\alpha-1})(1-acq^{\alpha-1})(1-adq^{\alpha-1})(1-abcdq^{\alpha-2})}{(1-abcdq^{2\alpha-2})(1-abcdq^{2\alpha-1})}\\ &\qquad+\frac{a}{q}\frac{(1-bcq^{\alpha})(1-bdq^{\alpha})(1-cdq^{\alpha})(1-q^{\alpha+1})}{(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha})}\\ &=z+z^{-1}-a-a^{-1}+A_{\alpha}+C_{\alpha}\\ &=2x-B_{\alpha} \end{align}
となることが確認できるので, 以下が得られる.