Askey-Wilson陪多項式3: Rahmanによる二重級数表示

$\displaystyle x=\frac{z+z^{-1}}2$とする. 前の記事 で, Ismail-RahmanによるAskey-Wilson陪多項式の明示式
\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}
を示した. 今回はRahmanによる以下の二重級数表示を示す.

\begin{align} r_n^{\alpha}(x)&=\frac{(abcdq^{2\alpha-1},q^{\alpha+1};q)_n}{(q,abcdq^{\alpha-1};q)_n}q^{-n\alpha}\sum_{k=0}^n\frac{(q^{-n},abcdq^{2\alpha+n-1},aq^{\alpha}z,aq^{\alpha}/z;q)_k}{(q^{\alpha+1},abq^{\alpha},acq^{\alpha},adq^{\alpha};q)_k}q^k\\ &\qquad\cdot\sum_{j=0}^k\frac{(q^{\alpha},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1};q)_j}{(q,abcdq^{2\alpha-2},aq^{\alpha}z,aq^{\alpha}/z;q)_j}q^j \end{align}

導出

Andrewsの恒等式 より,
\begin{align} &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)\\ &=\frac{(abcdq^{2\alpha+k-1},q^{-\alpha-n};q)_{n-k}}{(abcdq^{\alpha+k-1},q^{-n};q)_{n-k}}\sum_{j=0}^{n-k}\frac{(q^{k-n},abcdq^{2\alpha+n+k-1},q^{\alpha},aq^{k+1}/d;q)_j}{(q,abq^{\alpha+k},acq^{\alpha+k},q^{\alpha+k+1};q)_j}q^j\\ &\qquad\cdot\Q43{q^{-j},adq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1}}{abcdq^{2\alpha-2},adq^{\alpha+k},dq^{-j-k}/a}q\\ &=\frac{(abcdq^{2\alpha-1},q^{\alpha+1};q)_n}{(q,abcdq^{\alpha-1};q)_n}\frac{(q,abcdq^{\alpha-1};q)_k}{(abcdq^{2\alpha-1},q^{\alpha+1};q)_k}q^{\alpha(k-n)}\\ &\qquad\cdot\sum_{j=0}^{n-k}\frac{(q^{k-n},abcdq^{2\alpha+n+k-1},q^{\alpha},aq^{k+1}/d;q)_j}{(q,abq^{\alpha+k},acq^{\alpha+k},q^{\alpha+k+1};q)_j}q^j\\ &\qquad\cdot\Q43{q^{-j},adq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1}}{abcdq^{2\alpha-2},adq^{\alpha+k},dq^{-j-k}/a}q \end{align}
となる. ここで, Searsの変換公式 より
\begin{align} &\Q43{q^{-j},adq^{\alpha-1},bdq^{\alpha-1},cdq^{\alpha-1}}{abcdq^{2\alpha-2},adq^{\alpha+k},dq^{-j-k}/a}q\\ &=\frac{(a^2q^{\alpha+k},q^{k+1};q)_j}{(adq^{\alpha+k},aq^{k+1}/d;q)_j}\Q43{q^{-j},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1}}{abcdq^{2\alpha-2},a^2q^{\alpha+k},q^{-j-k}}q \end{align}
であるから, Ismail-Rahmanの表示に代入すると
\begin{align} r_n^{\alpha}(x)&=\frac{(abcdq^{2\alpha-1},q^{\alpha+1};q)_n}{(q,abcdq^{\alpha-1};q)_n}q^{-n\alpha}\sum_{k=0}^n\frac{(q^{-n},abcdq^{2\alpha+n-1},az,a/z;q)_k}{(q^{\alpha+1},abq^{\alpha},acq^{\alpha},adq^{\alpha};q)_k}q^{(\alpha+1)k}\\ &\qquad\cdot\sum_{j=0}^{n-k}\frac{(q^{k-n},abcdq^{2\alpha+n+k-1},q^{\alpha},a^2q^{\alpha+k},q^{k+1};q)_j}{(q,abq^{\alpha+k},acq^{\alpha+k},adq^{\alpha+k},q^{\alpha+k+1};q)_j}q^j\\ &\qquad\cdot\Q43{q^{-j},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1}}{abcdq^{2\alpha-2},a^2q^{\alpha+k},q^{-j-k}}q\\ &=\frac{(abcdq^{2\alpha-1},q^{\alpha+1};q)_n}{(q,abcdq^{\alpha-1};q)_n}q^{-n\alpha}\sum_{k=0}^n\sum_{j=0}^{n-k}\frac{(q^{\alpha};q)_j}{(q;q)_j}\frac{(az,a/z;q)_k}{(q,a^2q^{\alpha};q)_k}q^{j+(\alpha+1)k}\\ &\qquad\cdot\frac{(q^{-n},abcdq^{2\alpha+n-1},a^2q^{\alpha},q;q)_{j+k}}{(abq^{\alpha},acq^{\alpha},adq^{\alpha},q^{\alpha+1};q)_{j+k}}\\ &\qquad\cdot\Q43{q^{-j},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1}}{abcdq^{2\alpha-2},a^2q^{\alpha+k},q^{-j-k}}q\\ &=\frac{(abcdq^{2\alpha-1},q^{\alpha+1};q)_n}{(q,abcdq^{\alpha-1};q)_n}q^{-n\alpha}\sum_{m=0}^n\frac{(q^{-n},abcdq^{2\alpha+n-1},a^2q^{\alpha},q;q)_m}{(abq^{\alpha},acq^{\alpha},adq^{\alpha},q^{\alpha+1};q)_m}q^m\\ &\qquad\sum_{k=0}^m\frac{(q^{\alpha};q)_{m-k}}{(q;q)_{m-k}}\frac{(az,a/z;q)_k}{(q,a^2q^{\alpha};q)_k}q^{\alpha k}\Q43{q^{k-m},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1}}{abcdq^{2\alpha-2},a^2q^{\alpha+k},q^{-m}}q \end{align}
を得る. ここで,
\begin{align} &\sum_{k=0}^m\frac{(q^{\alpha};q)_{m-k}}{(q;q)_{m-k}}\frac{(az,a/z;q)_k}{(q,a^2q^{\alpha};q)_k}q^{\alpha k}\Q43{q^{k-m},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1}}{abcdq^{2\alpha-2},a^2q^{\alpha+k},q^{-m}}q\\ &=\frac{(q^{\alpha};q)_m}{(q;q)_m}\sum_{k=0}^m\frac{(az,a/z,q^{-m};q)_k}{(q,a^2q^{\alpha},q^{1-\alpha-m};q)_k}q^{k}\sum_{j=0}^m\frac{(q^{k-m},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1};q)_j}{(q,abcdq^{2\alpha-2},a^2q^{\alpha+k},q^{-m};q)_j}q^j\\ &=\frac{(q^{\alpha};q)_m}{(q;q)_m}\sum_{j=0}^m\frac{(abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1};q)_j}{(q,abcdq^{2\alpha-2},a^2q^{\alpha};q)_j}q^j\Q32{az,a/z,q^{j-m}}{a^2q^{\alpha+j},q^{1-\alpha-m}}q \end{align}
であり, $q$-Saalschützの和公式 より,
\begin{align} \Q32{az,a/z,q^{j-m}}{a^2q^{\alpha+j},q^{1-\alpha-m}}q&=\frac{(aq^{\alpha+j}/z,aq^{\alpha+j}z;q)_{m-j}}{(a^2q^{\alpha+j},q^{\alpha+j};q)_{m-j}}\\ &=\frac{(aq^{\alpha}z,aq^{\alpha}/z;q)_m}{(a^2q^{\alpha},q^{\alpha};q)_m}\frac{(a^2q^{\alpha},q^{\alpha};q)_j}{(aq^{\alpha}z,aq^{\alpha}/z;q)_j} \end{align}
であるから, これを代入すると
\begin{align} &\sum_{k=0}^m\frac{(q^{\alpha};q)_{m-k}}{(q;q)_{m-k}}\frac{(az,a/z;q)_k}{(q,a^2q^{\alpha};q)_k}q^{\alpha k}\Q43{q^{k-m},abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1}}{abcdq^{2\alpha-2},a^2q^{\alpha+k},q^{-m}}q\\ &=\frac{(aq^{\alpha}z,aq^{\alpha}/z;q)_m}{(a^2q^{\alpha},q;q)_m}\sum_{j=0}^m\frac{(abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1},q^{\alpha};q)_j}{(q,abcdq^{2\alpha-2},aq^{\alpha}z,aq^{\alpha}/z;q)_j}q^j \end{align}
となる. よって,
\begin{align} r_n^{\alpha}(x) &=\frac{(abcdq^{2\alpha-1},q^{\alpha+1};q)_n}{(q,abcdq^{\alpha-1};q)_n}q^{-n\alpha}\sum_{m=0}^n\frac{(q^{-n},abcdq^{2\alpha+n-1},a^2q^{\alpha},q;q)_m}{(abq^{\alpha},acq^{\alpha},adq^{\alpha},q^{\alpha+1};q)_m}q^m\\ &\qquad\cdot\frac{(aq^{\alpha}z,aq^{\alpha}/z;q)_m}{(a^2q^{\alpha},q;q)_m}\sum_{j=0}^m\frac{(abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1},q^{\alpha};q)_j}{(q,abcdq^{2\alpha-2},aq^{\alpha}z,aq^{\alpha}/z;q)_j}q^j\\ &=\frac{(abcdq^{2\alpha-1},q^{\alpha+1};q)_n}{(q,abcdq^{\alpha-1};q)_n}q^{-n\alpha}\sum_{m=0}^n\frac{(q^{-n},abcdq^{2\alpha+n-1},aq^{\alpha}z,aq^{\alpha}/z;q)_m}{(abq^{\alpha},acq^{\alpha},adq^{\alpha},q^{\alpha+1};q)_m}q^m\\ &\qquad\cdot\sum_{j=0}^m\frac{(abq^{\alpha-1},acq^{\alpha-1},adq^{\alpha-1},q^{\alpha};q)_j}{(q,abcdq^{2\alpha-2},aq^{\alpha}z,aq^{\alpha}/z;q)_j}q^j \end{align}
を得る. つまり以下が得られた.