Askey-Wilson陪多項式2: Ismail-Rahmanによる二重級数表示

前の記事 でAskey-Wilson陪多項式をAskey-Wilson関数で表す明示公式を得た. 前の記事の記法を引き続き用いることとして, それは$\displaystyle x=\frac{z+z^{-1}}2$
\begin{align} r_n^{\alpha}(x)&=\frac{S_{\alpha-1}(z)R_{n+\alpha}(z)-R_{\alpha-1}(z)S_{n+\alpha}(z)}{W_{\alpha}} \end{align}
と表されるものである. しかし, $R,S$は無限和で与えられるので, この表示からは多項式であることは明らかではない. 今回は, Askey-Wilson陪多項式を二重の有限和によって明示的に表すIsmail-Rahmanによる以下の公式を示す.

証明

\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} \end{align}
とする. 定義をあらためて
\begin{align} r_n^{\alpha}(x)&=r_n^{\alpha}(x;a,b,c,d)\\ &:=\sum_{k=0}^n\frac{(q^{-n},abcdq^{2\alpha+n-1},abcdq^{2\alpha-1},az,a/z;q)_k}{(q,abq^{\alpha},acq^{\alpha},adq^{\alpha},abcdq^{\alpha-1};q)_k}q^k\\ &\qquad\cdot W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k-1},q^{k-n};a^2) \end{align}
として, それがAskey-Wilson陪多項式の漸化式
\begin{align} 2xr_n^{\alpha}(x)=A_{n+\alpha}r_{n+1}^{\alpha}(x)+B_{n+\alpha}r_n^{\alpha}(x)+C_{n+\alpha}r_{n-1}^{\alpha}(x) \end{align}
を満たしていることを示せばよい. $n$に関する帰納法を用いる. $m< n$に対し
\begin{align} 2xr_m^{\alpha}(x)=A_{m+\alpha}r_{m+1}^{\alpha}(x)+B_{m+\alpha}r_m^{\alpha}(x)+C_{m+\alpha}r_{m-1}^{\alpha}(x) \end{align}
が成り立っているとする. このとき, $a,d$を入れ替えることによって,
\begin{align} \frac{(bdq^{\alpha},cdq^{\alpha};q)_m}{(abq^{\alpha},acq^{\alpha};q)_m}(a/d)^mr_m^{\alpha}(x;d,b,c,a) \end{align}
も$m< n$において全く同じ漸化式を満たしており, 初期値が等しいことが分かる. よって特に,
\begin{align} r_n^{\alpha}(x;a,b,c,d)=\frac{(bdq^{\alpha},cdq^{\alpha};q)_n}{(abq^{\alpha},acq^{\alpha};q)_n}(a/d)^nr_n^{\alpha}(x;d,b,c,a) \end{align}
が成り立つことが分かる. また,
\begin{align} &r_{n+1}^{\alpha}(x)-r_n^{\alpha}(x)\\ &=\sum_{0\leq k}\frac{(q^{-n-1},abcdq^{2\alpha+n},abcdq^{2\alpha-1},az,a/z;q)_k}{(q,abq^{\alpha},acq^{\alpha},adq^{\alpha},abcdq^{\alpha-1};q)_k}q^k\\ &\qquad\cdot W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k},q^{k-n-1};a^2)\\ &\qquad-\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n-1},abcdq^{2\alpha-1},az,a/z;q)_k}{(q,abq^{\alpha},acq^{\alpha},adq^{\alpha},abcdq^{\alpha-1};q)_k}q^k\\ &\qquad\cdot W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k-1},q^{k-n};a^2)\\ &=\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n};q)_{k-1}(abcdq^{2\alpha-1},az,a/z;q)_k}{(q,abq^{\alpha},acq^{\alpha},adq^{\alpha},abcdq^{\alpha-1};q)_k}q^k\\ &\qquad\cdot\bigg((1-q^{-n-1})(1-abcdq^{2\alpha+n+k-1})W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k},q^{k-n-1};a^2)\\ &\qquad\qquad-(1-q^{k-n-1})(1-abcdq^{2\alpha+n-1})W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k-1},q^{k-n};a^2)\bigg) \end{align}
となる. ここで,
\begin{align} &(1-q^{-n-1})(1-abcdq^{2\alpha+n+k-1})\frac{(abcdq^{2\alpha+n+k},q^{k-n-1};q)_j}{(q^{-n-1},abcdq^{2\alpha+n};q)_j}\\ &\qquad-(1-q^{k-n-1})(1-abcdq^{2\alpha+n-1})\frac{(abcdq^{2\alpha+n+k-1},q^{k-n};q)_j}{(q^{-n},abcdq^{2\alpha+n-1};q)_j}\\ &=\frac{(abcdq^{2\alpha+n+k-1};q)_{j+1}(q^{k-n-1};q)_j}{(q^{-n};q)_{j-1}(abcdq^{2\alpha+n};q)_j}-\frac{(abcdq^{2\alpha+n+k-1};q)_j(q^{k-n-1};q)_{j+1}}{(q^{-n};q)_j(abcdq^{2\alpha+n};q)_{j-1}}\\ &=\frac{(abcdq^{2\alpha+n+k-1},q^{k-n-1};q)_j}{(q^{-n},abcdq^{2\alpha+n};q)_j}((1-q^{j-n-1})(1-abcdq^{2\alpha+n+k+j-1})-(1-q^{j+k-n-1})(1-abcdq^{2\alpha+n+j-1}))\\ &=\frac{(abcdq^{2\alpha+n+k-1},q^{k-n-1};q)_j}{(q^{-n},abcdq^{2\alpha+n};q)_j}(1-q^k)(abcdq^{2\alpha+n+j-1}-q^{j-n-1})\\ &=-q^{-n-1}(1-abcdq^{2\alpha+2n})(1-q^k)\frac{(abcdq^{2\alpha+n+k-1},q^{k-n-1};q)_j}{(q^{-n},abcdq^{2\alpha+n};q)_j}q^j \end{align}
となることから,

\begin{align} &(1-q^{-n-1})(1-abcdq^{2\alpha+n+k-1})W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k},q^{k-n-1};a^2)\\ &\qquad-(1-q^{k-n-1})(1-abcdq^{2\alpha+n-1})W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k-1},q^{k-n};a^2)\\ &=-q^{-n-1}(1-abcdq^{2\alpha+2n})(1-q^k)W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k-1},q^{k-n-1};a^2q) \end{align}
を得る. これを代入すると
\begin{align} &r_{n+1}^{\alpha}(x)-r_n^{\alpha}(x)\\ &=-q^{-n-1}(1-abcdq^{2\alpha+2n})\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n};q)_{k-1}(abcdq^{2\alpha-1},az,a/z;q)_k}{(q;q)_{k-1}(abq^{\alpha},acq^{\alpha},adq^{\alpha},abcdq^{\alpha-1};q)_k}q^k\\ &\qquad\cdot W(abcdq^{2\alpha+k-2};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+1},abcdq^{2\alpha+n+k-1},q^{k-n-1};a^2q)\\ &=-\frac{q^{-n}(1-az)(1-a/z)(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha+2n})}{(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}\\ &\qquad\cdot\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n},abcdq^{2\alpha},aqz,aq/z;q)_{k}}{(q,abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1},abcdq^{\alpha};q)_k}q^k\\ &\qquad\cdot W(abcdq^{2\alpha+k-1};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+2},abcdq^{2\alpha+n+k},q^{k-n};a^2q)\\ &=\frac{aq^{-n}(z+z^{-1}-a-a^{-1})(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha+2n})}{(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}\\ &\qquad\cdot\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n},abcdq^{2\alpha},aqz,aq/z;q)_{k}}{(q,abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1},abcdq^{\alpha};q)_k}q^k\\ &\qquad\cdot W(abcdq^{2\alpha+k-1};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+2},abcdq^{2\alpha+n+k},q^{k-n};a^2q) \end{align}
を得る. ここで, Andrewsの恒等式 より,
\begin{align} &W(abcdq^{2\alpha+k-1};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+2},abcdq^{2\alpha+n+k},q^{k-n};a^2q)\\ &=\frac{(abcdq^{2\alpha+k},adq^{k+1};q)_{n-k}}{(abcdq^{\alpha+k},adq^{\alpha+k+1};q)_{n-k}}\sum_{j=0}^{n-k}\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},aq^{k+2}/d;q)_j}{(q,abq^{\alpha+k+1},acq^{\alpha+k+1},q^{-n}/ad;q)_j}q^j\Q43{q^{-n-k-2},bdq^{\alpha-1},cdq^{\alpha-1},q^{-j}}{abcdq^{2\alpha-2},q^{-n},dq^{-j-k-1}/a}{q} \end{align}
であり, Searの変換公式 より
\begin{align} &\Q43{q^{-n-k-2},bdq^{\alpha-1},cdq^{\alpha-1},q^{-j}}{abcdq^{2\alpha-2},q^{-n},dq^{-j-k-1}/a}{q}\\ &=\frac{(abq^{\alpha+k+1},q^{1-\alpha-n}/bd;q)_j}{(q^{-n},aq^{k+2}/d;q)_j}\Q43{abcdq^{2\alpha+n+k},abq^{\alpha-1},bdq^{\alpha-1},q^{-j}}{abcdq^{2\alpha-2},abq^{\alpha+k+1},bdq^{\alpha+n-j}}q \end{align}
となるから, これを代入すると,

\begin{align} S_n&:=\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n},abcdq^{2\alpha},aqz,aq/z;q)_{k}}{(q,abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1},abcdq^{\alpha};q)_k}q^k\\ &\qquad\cdot W(abcdq^{2\alpha+k-1};q^{\alpha},bcq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1},q^{k+2},abcdq^{2\alpha+n+k},q^{k-n};a^2q)\\ &=\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n},abcdq^{2\alpha},aqz,aq/z;q)_{k}}{(q,abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1},abcdq^{\alpha};q)_k}q^k\frac{(abcdq^{2\alpha+k},adq^{k+1};q)_{n-k}}{(abcdq^{\alpha+k},adq^{\alpha+k+1};q)_{n-k}}\\ &\qquad\cdot\sum_{j=0}^{n-k}\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},aq^{k+2}/d;q)_j}{(q,abq^{\alpha+k+1},acq^{\alpha+k+1},q^{-n}/ad;q)_j}q^j\frac{(abq^{\alpha+k+1},q^{1-\alpha-n}/bd;q)_j}{(q^{-n},aq^{k+2}/d;q)_j}\\ &\qquad\cdot\Q43{abcdq^{2\alpha+n+k},abq^{\alpha-1},bdq^{\alpha-1},q^{-j}}{abcdq^{2\alpha-2},abq^{\alpha+k+1},bdq^{\alpha+n-j}}q\\ &=\frac{(abcdq^{2\alpha},adq;q)_n}{(abcdq^{\alpha},adq^{\alpha+1};q)_n}\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n},aqz,aq/z;q)_{k}}{(q,abq^{\alpha+1},acq^{\alpha+1},adq;q)_k}q^k\\ &\qquad\cdot\sum_{j=0}^{n}\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},q^{1-\alpha-n}/bd;q)_j}{(q,acq^{\alpha+k+1},q^{-n}/ad,q^{-n};q)_j}q^j\sum_{i=0}^j\frac{(abcdq^{2\alpha+n+k},abq^{\alpha-1},bdq^{\alpha-1},q^{-j};q)_i}{(q,abcdq^{2\alpha-2},abq^{\alpha+k+1},bdq^{\alpha+n-j};q)_i}q^i\\ &=\frac{(abcdq^{2\alpha},adq;q)_n}{(abcdq^{\alpha},adq^{\alpha+1};q)_n}\sum_{j=0}^{n}\frac{(q^{-n},q^{\alpha},bcq^{\alpha-1},q^{1-\alpha-n}/bd;q)_j}{(q,acq^{\alpha+1},q^{-n}/ad,q^{-n};q)_j}q^j\sum_{i=0}^j\frac{(abcdq^{2\alpha+n},abq^{\alpha-1},bdq^{\alpha-1},q^{-j};q)_i}{(q,abcdq^{2\alpha-2},abq^{\alpha+1},bdq^{\alpha+n-j};q)_i}q^i\\ &\qquad\cdot\Q43{q^{j-n},abcdq^{2\alpha+n+i},aqz,aq/z}{abq^{\alpha+i+1},acq^{\alpha+j+1},adq}q \end{align}
を得る. ここで, Searsの変換公式 より,
\begin{align} &\Q43{q^{j-n},abcdq^{2\alpha+n+i},aqz,aq/z}{abq^{\alpha+i+1},acq^{\alpha+j+1},adq}q\\ &=\frac{(bdq^{\alpha+i},cdq^{\alpha+j};q)_{n-j}}{(abq^{\alpha+i+1},acq^{\alpha+j+1};q)_{n-j}}(aq/d)^{n-j}\Q43{q^{j-n},abcdq^{2\alpha+n+i},dz,d/z}{adq,bdq^{\alpha+i},cdq^{\alpha+j}}q \end{align}
となる. この係数は
\begin{align} &\frac{(bdq^{\alpha+i},cdq^{\alpha+j};q)_{n-j}}{(abq^{\alpha+i+1},acq^{\alpha+j+1};q)_{n-j}}(abcdq^{2\alpha+n+i})^{n-j}\\ &=\frac{(cdq^{\alpha};q)_n}{(acq^{\alpha+1};q)_n}\frac{(bdq^{\alpha};q)_{n-j+i}(acq^{\alpha+1};q)_j(abq^{\alpha+1};q)_i}{(abq^{\alpha+1};q)_{n-j+i}(cdq^{\alpha};q)_j(bdq^{\alpha};q)_i}(aq/d)^{n-j}\\ &=\frac{(bdq^{\alpha},cdq^{\alpha};q)_n}{(abq^{\alpha+1},acq^{\alpha+1};q)_n}\frac{(q^{-\alpha-n}/ab;q)_{j-i}(acq^{\alpha+1};q)_j(abq^{\alpha+1};q)_i}{(q^{1-\alpha-n}/bd;q)_{j-i}(cdq^{\alpha};q)_j(bdq^{\alpha};q)_i}(aq/d)^{n-i}\\ &=\frac{(bdq^{\alpha},cdq^{\alpha};q)_n}{(abq^{\alpha+1},acq^{\alpha+1};q)_n}(aq/d)^{n}\frac{(acq^{\alpha+1},q^{-\alpha-n}/ab;q)_j(abq^{\alpha+1},bdq^{\alpha+n-j};q)_i}{(cdq^{\alpha},q^{1-\alpha-n}/bd;q)_j(bdq^{\alpha},abq^{1+\alpha+n-j};q)_i} \end{align}
と表される. よって, これを代入すると,
\begin{align} S_n&=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\\ &\qquad\cdot\sum_{j=0}^{n}\frac{(q^{-n},q^{\alpha},bcq^{\alpha-1},q^{1-\alpha-n}/bd;q)_j}{(q,acq^{\alpha+1},q^{-n}/ad,q^{-n};q)_j}q^j\frac{(acq^{\alpha+1},q^{-\alpha-n}/ab;q)_j}{(cdq^{\alpha},q^{1-\alpha-n}/bd;q)_j}\\ &\qquad\cdot\sum_{i=0}^j\frac{(abcdq^{2\alpha+n},abq^{\alpha-1},bdq^{\alpha-1},q^{-j};q)_i}{(q,abcdq^{2\alpha-2},abq^{\alpha+1},bdq^{\alpha+n-j};q)_i}q^i\frac{(abq^{\alpha+1},bdq^{\alpha+n-j};q)_i}{(bdq^{\alpha},abq^{1+\alpha+n-j};q)_i}\\ &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\\ &\qquad\cdot\sum_{j=0}^{n}\frac{(q^{-n},q^{\alpha},bcq^{\alpha-1},q^{1-\alpha-n}/bd;q)_j}{(q,acq^{\alpha+1},q^{-n}/ad,q^{-n};q)_j}q^j\frac{(acq^{\alpha+1},q^{-\alpha-n}/ab;q)_j}{(cdq^{\alpha},q^{1-\alpha-n}/bd;q)_j}\\ &\qquad\cdot\sum_{i=0}^j\frac{(abcdq^{2\alpha+n},abq^{\alpha-1},bdq^{\alpha-1},q^{-j};q)_i}{(q,abcdq^{2\alpha-2},abq^{\alpha+1},bdq^{\alpha+n-j};q)_i}q^i\frac{(abq^{\alpha+1},bdq^{\alpha+n-j};q)_i}{(bdq^{\alpha},abq^{1+\alpha+n-j};q)_i}\Q43{q^{j-n},abcdq^{2\alpha+n+i},dz,d/z}{adq,bdq^{\alpha+i},cdq^{\alpha+j}}q\\ &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{j=0}^{n}\frac{(q^{-n},q^{\alpha},bcq^{\alpha-1},q^{-\alpha-n}/ab;q)_j}{(q,q^{-n}/ad,q^{-n},cdq^{\alpha};q)_j}q^j\\ &\qquad\cdot\sum_{i=0}^j\frac{(abcdq^{2\alpha+n},abq^{\alpha-1},bdq^{\alpha-1},q^{-j};q)_i}{(q,abcdq^{2\alpha-2},bdq^{\alpha},abq^{1+\alpha+n-j};q)_i}q^i\sum_{k=0}^{n}\frac{(q^{j-n},abcdq^{2\alpha+n+i},dz,d/z;q)_k}{(q,adq,bdq^{\alpha+i},cdq^{\alpha+j};q)_k}q^k\\ &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q,adq,bdq^{\alpha},cdq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{j=0}^{n}\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},q^{-\alpha-n}/ab;q)_j}{(q,q^{-n}/ad,q^{-n},cdq^{\alpha+k};q)_j}q^j\Q43{abcdq^{2\alpha+n+k},abq^{\alpha-1},bdq^{\alpha-1},q^{-j}}{abcdq^{2\alpha-2},bdq^{\alpha+k},abq^{1+\alpha+n-j}}q \end{align}
ここで, Watsonの変換公式 より
\begin{align} &\Q43{abcdq^{2\alpha+n+k},abq^{\alpha-1},bdq^{\alpha-1},q^{-j}}{abcdq^{2\alpha-2},bdq^{\alpha+k},abq^{1+\alpha+n-j}}q\\ &=\frac{(cdq^{\alpha+k},q^{-n-1};q)_j}{(abcdq^{2\alpha+k-1},q^{-\alpha-n}/ab;q)_j}\\ &\qquad\cdot\Q87{abcdq^{2\alpha+k-2},q\sqrt{abcdq^{2\alpha+k-2}},-q\sqrt{abcdq^{2\alpha+k-2}},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k},q^{-j}}{\sqrt{abcdq^{2\alpha+k-2}},-\sqrt{abcdq^{2\alpha+k-2}},abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{-n-1},abcdq^{2\alpha+j+k-1}}{\frac{dq^{j-n-1}}{a}} \end{align}
であるから, これを代入して,
\begin{align} S_n &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q,adq,bdq^{\alpha},cdq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{j=0}^{n}\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},q^{-\alpha-n}/ab;q)_j}{(q,q^{-n}/ad,q^{-n},cdq^{\alpha+k};q)_j}q^j\frac{(cdq^{\alpha+k},q^{-n-1};q)_j}{(abcdq^{2\alpha+k-1},q^{-\alpha-n}/ab;q)_j}\\ &\qquad\cdot\Q87{abcdq^{2\alpha+k-2},q\sqrt{abcdq^{2\alpha+k-2}},-q\sqrt{abcdq^{2\alpha+k-2}},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k},q^{-j}}{\sqrt{abcdq^{2\alpha+k-2}},-\sqrt{abcdq^{2\alpha+k-2}},abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{-n-1},abcdq^{2\alpha+j+k-1}}{\frac{dq^{j-n-1}}{a}}\\ &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q,adq,bdq^{\alpha},cdq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{j=0}^{n}\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},q^{-n-1};q)_j}{(q,q^{-n}/ad,q^{-n},abcdq^{2\alpha+k-1};q)_j}q^j\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k},q^{-j};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{-n-1},abcdq^{2\alpha+j+k-1};q)_i}\left(\frac{dq^{j-n-1}}{a}\right)^i\\ &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q,adq,bdq^{\alpha},cdq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{-n-1};q)_i}\left(-\frac{dq^{-n-1}}{a}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{j=0}^{n}\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},q^{-n-1};q)_j}{(q^{-n}/ad,q^{-n};q)_j(abcdq^{2\alpha+k-1};q)_{j+i}(q;q)_{j-i}}q^j\\ &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q,adq,bdq^{\alpha},cdq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{-n-1};q)_i}\left(-\frac{dq^{-n-1}}{a}\right)^iq^{\binom i2}\\ &\qquad\cdot\frac{(q^{k-n},q^{\alpha},bcq^{\alpha-1},q^{-n-1};q)_i}{(q^{-n}/ad,q^{-n};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}q^i\Q43{q^{i+k-n},q^{\alpha+i},bcq^{\alpha+i-1},q^{i-n-1}}{q^{i-n}/ad,q^{i-n},abcdq^{2\alpha+k-1+2i}}q\\ &=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q,adq,bdq^{\alpha},cdq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k},q^{k-n},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{-n}/ad,q^{-n};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-\frac{dq^{-n}}{a}\right)^iq^{\binom i2}\\ &\qquad\cdot\Q43{q^{i+k-n},q^{\alpha+i},bcq^{\alpha+i-1},q^{i-n-1}}{q^{i-n}/ad,q^{i-n},abcdq^{2\alpha+k-1+2i}}q \end{align}
ここで, Searsの変換公式 より
\begin{align} &\Q43{q^{i+k-n},q^{\alpha+i},bcq^{\alpha+i-1},q^{i-n-1}}{q^{i-n}/ad,q^{i-n},abcdq^{2\alpha+k-1+2i}}q\\ &=\frac{(q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_{n-k-i}}{(adq^{k+1},q^{k+1};q)_{n-k-i}}(q^{\alpha+i})^{i+k-n}\Q43{q^{i+k-n},q^{\alpha+i},adq^{\alpha+i+k},abcdq^{2\alpha+n+k+i}}{abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1}}q\\ &=\frac{(q^{\alpha+1},adq^{\alpha+1};q)_n(adq,q;q)_k}{(adq,q;q)_{n-i}(q^{\alpha+1},adq^{\alpha+1};q)_{k+i}}(q^{\alpha+i})^{i+k-n}\Q43{q^{i+k-n},q^{\alpha+i},adq^{\alpha+i+k},abcdq^{2\alpha+n+k+i}}{abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1}}q\\ &=\frac{(q^{\alpha+1},adq^{\alpha+1};q)_n(adq,q;q)_k(q^{-n},q^{-n}/ad;q)_i}{(adq,q;q)_{n}(q^{\alpha+1},adq^{\alpha+1};q)_{k+i}}q^{\alpha(k-n)}(adq^{\alpha+n+k+1})^i\\ &\qquad\cdot\Q43{q^{i+k-n},q^{\alpha+i},adq^{\alpha+i+k},abcdq^{2\alpha+n+k+i}}{abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1}}q \end{align}
を得る. よって, これを代入すると,
\begin{align} S_n&=\frac{(abcdq^{2\alpha},adq,bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},adq^{\alpha+1};q)_n}(aq/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q,adq,bdq^{\alpha},cdq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k},q^{k-n},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{-n}/ad,q^{-n};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-\frac{dq^{-n}}{a}\right)^iq^{\binom i2}\\ &\qquad\cdot\frac{(q^{\alpha+1},adq^{\alpha+1};q)_n(adq,q;q)_k(q^{-n},q^{-n}/ad;q)_i}{(adq,q;q)_{n}(q^{\alpha+1},adq^{\alpha+1};q)_{k+i}}q^{\alpha(k-n)}(adq^{\alpha+n+k+1})^i\\ &\qquad\cdot\Q43{q^{i+k-n},q^{\alpha+i},adq^{\alpha+i+k},abcdq^{2\alpha+n+k+i}}{abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1}}q\\ &=\frac{(abcdq^{2\alpha},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},q;q)_n}(aq^{1-\alpha}/d)^{n}\sum_{k=0}^{n}\frac{(q^{-n},abcdq^{2\alpha+n},dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},abcdq^{2\alpha+n+k},q^{k-n},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\Q43{q^{i+k-n},q^{\alpha+i},adq^{\alpha+i+k},abcdq^{2\alpha+n+k+i}}{abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1}}q\\ &=\frac{(abcdq^{2\alpha},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},q;q)_n}(aq^{1-\alpha}/d)^{n}\sum_{k=0}^{n}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i},adq^{\alpha+i+k};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_j}q^j(q^{-n},abcdq^{2\alpha+n};q)_{i+j+k} \end{align}
となる. よって,
\begin{align} &r_{n+1}^{\alpha}(x)-r_n^{\alpha}(x)\\ &=\frac{aq^{-n}(z+z^{-1}-a-a^{-1})(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha+2n})}{(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}S_n\\ &=\frac{aq^{-n}(z+z^{-1}-a-a^{-1})(1-abcdq^{2\alpha-1})(1-abcdq^{2\alpha+2n})}{(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}\\ &\qquad\cdot\frac{(abcdq^{2\alpha},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},q;q)_n}(aq^{1-\alpha}/d)^{n}\sum_{k=0}^{n}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i},adq^{\alpha+i+k};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_j}q^j(q^{-n},abcdq^{2\alpha+n};q)_{i+j+k} \end{align}
となる. これを用いると,
\begin{align} &A_{n+\alpha}(r_{n+1}^{\alpha}(x)-r_n^{\alpha}(x))-C_{n+\alpha}(r_{n}^{\alpha}(x)-r_{n-1}^{\alpha}(x))\\ &=\frac{q^{-n}(z+z^{-1}-a-a^{-1})(1-abcdq^{2\alpha-1})(1-abq^{\alpha+n})(1-acq^{\alpha+n})(1-adq^{\alpha+n})(1-abcdq^{\alpha+n-1})}{(1-abcdq^{2\alpha+2n-1})(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}\\ &\qquad\cdot\frac{(abcdq^{2\alpha},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},q;q)_n}(aq^{1-\alpha}/d)^{n}\sum_{0\leq k}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i},adq^{\alpha+i+k};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_j}q^j(q^{-n},abcdq^{2\alpha+n};q)_{i+j+k}\\ &-\frac{a^2q^{1-n}(z+z^{-1}-a-a^{-1})(1-abcdq^{2\alpha-1})(1-bcq^{\alpha+n-1})(1-bdq^{\alpha+n-1})(1-cdq^{\alpha+n-1})(1-q^{\alpha+n})}{(1-abcdq^{2\alpha+2n-1})(1-abq^{\alpha})(1-acq^{\alpha})(1-adq^{\alpha})(1-abcdq^{\alpha-1})}\\ &\qquad\cdot\frac{(abcdq^{2\alpha},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_{n-1}}{(abcdq^{\alpha},abq^{\alpha+1},acq^{\alpha+1},q;q)_{n-1}}(aq^{1-\alpha}/d)^{n-1}\sum_{0\leq k}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i},adq^{\alpha+i+k};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_j}q^j(q^{1-n},abcdq^{2\alpha+n-1};q)_{i+j+k}\\ &=\frac{q^{-n}(z+z^{-1}-a-a^{-1})(1-abcdq^{2\alpha+n-1})(1-adq^{\alpha+n})}{(1-abcdq^{2\alpha+2n-1})(1-adq^{\alpha})}\\ &\qquad\cdot\frac{(abcdq^{2\alpha-1},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha-1},abq^{\alpha},acq^{\alpha},q;q)_n}(aq^{1-\alpha}/d)^{n}\sum_{0\leq k}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i},adq^{\alpha+i+k};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_j}q^j(q^{-n},abcdq^{2\alpha+n};q)_{i+j+k}\\ &-\frac{adq^{\alpha-n}(z+z^{-1}-a-a^{-1})(1-bcq^{\alpha+n-1})(1-q^n)}{(1-abcdq^{2\alpha+2n-1})(1-adq^{\alpha})}\\ &\qquad\cdot\frac{(abcdq^{2\alpha-1},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_{n}}{(abcdq^{\alpha-1},abq^{\alpha},acq^{\alpha},q;q)_{n}}(aq^{1-\alpha}/d)^{n}\sum_{0\leq k}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i},adq^{\alpha+i+k};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_j}q^j(q^{1-n},abcdq^{2\alpha+n-1};q)_{i+j+k}\\ &=\frac{z+z^{-1}-a-a^{-1}}{(1-abcdq^{2\alpha+2n-1})(1-adq^{\alpha})}\frac{(abcdq^{2\alpha-1},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha-1},abq^{\alpha},acq^{\alpha},q;q)_n}(aq^{-\alpha}/d)^{n}\sum_{0\leq k}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},cdq^{\alpha+k},bdq^{\alpha+k},q^{\alpha+k+1},adq^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i},adq^{\alpha+i+k};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1},adq^{\alpha+i+k+1};q)_j}q^j\\ &\qquad\cdot((1-abcdq^{2\alpha+n-1})(1-adq^{\alpha+n})(q^{-n},abcdq^{2\alpha+n};q)_{i+j+k}-adq^{\alpha}(1-bcq^{\alpha+n-1})(1-q^n)(q^{1-n},abcdq^{2\alpha+n-1};q)_{i+j+k}) \end{align}
となる. ここで,
\begin{align} &(1-abcdq^{2\alpha+n-1})(1-adq^{\alpha+n})(q^{-n},abcdq^{2\alpha+n};q)_{i+j+k}-adq^{\alpha}(1-bcq^{\alpha+n-1})(1-q^n)(q^{1-n},abcdq^{2\alpha+n-1};q)_{i+j+k}\\ &=((1-abcdq^{2\alpha+n+i+j+k-1})(1-adq^{\alpha+n})(1-q^{-n})-adq^{\alpha}(1-bcq^{\alpha+n-1})(1-q^n)(1-q^{i+j+k-n}))\\ &\qquad\cdot(q^{1-n};q)_{i+j+k-1}(abcdq^{2\alpha+n-1};q)_{i+j+k}\\ &=(1-q^{-n})((1-abcdq^{2\alpha+n+i+j+k-1})(1-adq^{\alpha+n})+adq^{\alpha+n}(1-bcq^{\alpha+n-1})(1-q^{i+j+k-n}))\\ &\qquad\cdot(q^{1-n};q)_{i+j+k-1}(abcdq^{2\alpha+n-1};q)_{i+j+k}\\ &=((1+a^2bcd^2q^{3\alpha+2n+i+j+k-1})-(abcdq^{2\alpha+2n-1}+adq^{\alpha+i+j+k}))\\ &\qquad\cdot(q^{-n},abcdq^{2\alpha+n-1};q)_{i+j+k}\\ &=(1-adq^{\alpha+i+j+k})(1-abcdq^{2\alpha+2n-1})(q^{1-n},abcdq^{2\alpha+n-1};q)_{i+j+k} \end{align}
となるから, これを代入すると,
\begin{align} &A_{n+\alpha}(r_{n+1}^{\alpha}(x)-r_n^{\alpha}(x))-C_{n+\alpha}(r_{n}^{\alpha}(x)-r_{n-1}^{\alpha}(x))\\ &=\frac{(2x-a-a^{-1})(abcdq^{2\alpha-1},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha-1},abq^{\alpha},acq^{\alpha},q;q)_n}(aq^{-\alpha}/d)^{n}\sum_{0\leq k}\frac{(dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},adq^{\alpha+k},bdq^{\alpha+k},cdq^{\alpha+k},q^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\sum_{0\leq j}\frac{(q^{\alpha+i};q)_j}{(q,abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1};q)_j}q^j(q^{-n},abcdq^{2\alpha+n-1};q)_{i+j+k}\\ &=\frac{(2x-a-a^{-1})(abcdq^{2\alpha-1},q^{\alpha+1},bdq^{\alpha},cdq^{\alpha};q)_n}{(abcdq^{\alpha-1},abq^{\alpha},acq^{\alpha},q;q)_n}(aq^{-\alpha}/d)^{n}\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n-1},dz,d/z;q)_k}{(q^{\alpha+1},adq^{\alpha},bdq^{\alpha},cdq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},q^{\alpha},bcq^{\alpha-1},q^{k-n},abcdq^{2\alpha+n+k-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},adq^{\alpha+k},bdq^{\alpha+k},cdq^{\alpha+k},q^{\alpha+k+1};q)_i(abcdq^{2\alpha+k-1};q)_{2i}}\left(-d^2q^{\alpha+k+1}\right)^iq^{\binom i2}\\ &\qquad\cdot\Q32{q^{\alpha+i},q^{i+k-n},abcdq^{2\alpha+n+i+k-1}}{abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1}}q \end{align}
となる. ここで, $q$-Saalschützの和公式 より,
\begin{align} &\Q32{q^{\alpha+i},q^{i+k-n},abcdq^{2\alpha+n+i+k-1}}{abcdq^{2\alpha+k-1+2i},q^{\alpha+i+k+1}}q\\ &=\frac{(q^{k+1},abcdq^{\alpha+k+i-1};q)_{n-k-i}}{(q^{\alpha+i+k+1},abcdq^{2\alpha+k-1+2i};q)_{n-k-i}}(q^{\alpha+i})^{n-k-i}\\ &=\frac{(abcdq^{\alpha-1};q)_n(q;q)_{n-i}(q^{\alpha+1};q)_{i+k}(abcdq^{2\alpha-1};q)_{2i+k}}{(q^{\alpha+1};q)_n(q;q)_k(abcdq^{\alpha-1};q)_{i+k}(abcdq^{2\alpha-1};q)_{n+i}}(q^{\alpha+i})^{n-k-i}\\ &=\frac{(q,abcdq^{\alpha-1};q)_n}{(q^{\alpha+1},abcdq^{2\alpha-1};q)_n}q^{\alpha(n-k)}\frac{(q^{\alpha+1},abcdq^{2\alpha-1};q)_k}{(q,abcdq^{\alpha-1};q)_k}\frac{(q^{\alpha+k+1};q)_{i}(abcdq^{2\alpha+k-1};q)_{2i}}{(abcdq^{\alpha+k-1},abcdq^{2\alpha+n-1},q^{-n};q)_i}(-1)^iq^{-\binom{i+1}2-i(k+\alpha)}\\ \end{align}
となるから, これを代入して, 帰納法の仮定より,
\begin{align} &A_{n+\alpha}(r_{n+1}^{\alpha}(x)-r_n^{\alpha}(x))-C_{n+\alpha}(r_{n}^{\alpha}(x)-r_{n-1}^{\alpha}(x))\\ &=\frac{(2x-a-a^{-1})(bdq^{\alpha},cdq^{\alpha};q)_n}{(abq^{\alpha},acq^{\alpha};q)_n}(a/d)^{n}\sum_{0\leq k}\frac{(q^{-n},abcdq^{2\alpha+n-1},abcdq^{2\alpha-1},dz,d/z;q)_k}{(q,adq^{\alpha},bdq^{\alpha},cdq^{\alpha},abcdq^{\alpha-1};q)_k}q^{k}\\ &\qquad\cdot\sum_{0\leq i}\frac{(1-abcdq^{2\alpha+k-2+2i})(abcdq^{2\alpha+k-2},q^{k+1},abq^{\alpha-1},acq^{\alpha-1},bcq^{\alpha-1},q^{\alpha},q^{k-n},abcdq^{2\alpha+n+k-1};q)_i}{(1-abcdq^{2\alpha+k-2})(q,abcdq^{2\alpha-2},adq^{\alpha+k},bdq^{\alpha+k},cdq^{\alpha+k},abcdq^{\alpha+k-1},abcdq^{2\alpha+n-1},q^{-n};q)_i}d^{2i}\\ &=\frac{(2x-a-a^{-1})(bdq^{\alpha},cdq^{\alpha};q)_n}{(abq^{\alpha},acq^{\alpha};q)_n}(a/d)^{n}r_n^{\alpha}(x;d,b,c,a) \end{align}
を得る. 最初に示した等式
\begin{align} \frac{(bdq^{\alpha},cdq^{\alpha};q)_n}{(abq^{\alpha},acq^{\alpha};q)_n}(a/d)^{n}r_n^{\alpha}(x;d,b,c,a)=r_n^{\alpha}(x;a,b,c,d)=r_n^{\alpha}(x) \end{align}
であることを用いれば
\begin{align} &A_{n+\alpha}(r_{n+1}^{\alpha}(x)-r_n^{\alpha}(x))-C_{n+\alpha}(r_{n}^{\alpha}(x)-r_{n-1}^{\alpha}(x))=(2x-a-a^{-1})r_n^{\alpha}(x) \end{align}
つまり,
\begin{align} 2xr_n^{\alpha}(x)=A_{n+\alpha}r_{n+1}^{\alpha}(x)+B_{n+\alpha}r_n^{\alpha}(x)+C_{n+\alpha}r_{n-1}^{\alpha}(x) \end{align}
となって示すべき等式が得られた.