Askey-Wilson多項式の積公式
Askey-Wilson多項式は
\begin{align}
p_n(x;a,b,c,d|q):=a^{-n}(ab,ac,ad;q)_n\Q43{q^{-n},abcdq^{n-1},ae^{i\theta},ae^{-i\theta}}{ab,ac,ad}q
\end{align}
によって定義される. 今回は以下のような積公式を示す.
$x=\cos\theta,y=\cos\phi$とするとき,
\begin{align}
&p_n(x;a,b,c,d|q)p_n(y;a,b,c,d|q)\\
&=(ab,ac,ad,ad,bd,cd;q)_n(ad)^{-n}\\
&\qquad\cdot\sum_{m=0}^n\frac{(q^{-n},abcdq^{n-1},de^{i\theta},de^{-i\theta},de^{i\phi},de^{-i\phi};q)_m}{(q,ad,ad,bd,cd,d/a;q)_m}q^m\\
&\qquad\Q{10}9{aq^{-m}/d,q\sqrt{aq^{-m}/d},-q\sqrt{aq^{-m}/d},q^{1-m}/bd,q^{1-m}/cd,q^{-m},ae^{i\theta},ae^{-i\theta},ae^{i\phi},ae^{-i\phi}}{\sqrt{aq^{-m}/d},-\sqrt{aq^{-m}/d},ab,ac,aq/d,e^{-i\theta}q^{1-m}/d,e^{i\theta}q^{1-m}/d,e^{-i\phi}q^{1-m}/d,e^{i\phi}q^{1-m}/d}{\frac{bcq}{ad}}
\end{align}
が成り立つ.
Terminating balanced${}_4\phi_3$の積公式
\begin{align}
&\Q43{q^{-n},aq^n,b_1,b_2}{b,b_3,ab_1b_2q/bb_3}q\Q43{q^{-n},aq^n,c_1,c_2}{aq/b,aq/b_3,bb_3c_1c_2/aq}q\\
&=\sum_{0\leq m}\frac{(q^{-n},aq^n,ab_1q/bb_3,ab_2q/bb_3,c_1,c_2;q)_{m}}{(q,aq/bb_3,ab_1b_2q/bb_3,aq/b,aq/b_3,bb_3c_1c_2/aq;q)_m}q^{m}\\
&\qquad\cdot\Q{10}{9}{bb_3q^{-m-1}/a,q\sqrt{bb_3q^{-m-1}/a},-q\sqrt{bb_3q^{-m-1}/a},b_1,b_2,b_3q^{-m}/a,bq^{-m}/a,bb_3c_1/aq,bb_3c_2/aq,q^{-m}}{\sqrt{bb_3q^{-m-1}/a},-\sqrt{bb_3q^{-m-1}/a},bb_3q^{-m}/ab_1,bb_3q^{-m}/ab_2,b_3,b,q^{1-m}/c_1,q^{1-m}/c_2,bb_3/a}{\frac{aq^2}{b_1b_2c_1c_2}}
\end{align}
において, $a\mapsto abcd/q,b\mapsto ab,b_1\mapsto ae^{i\theta},b_2\mapsto ae^{-i\theta},b_3\mapsto ac,c_1\mapsto de^{i\phi},c_2\mapsto de^{-i\phi}$とすると,
\begin{align}
&\Q43{q^{-n},abcdq^{n-1},ae^{i\theta},ae^{-i\theta}}{ab,ac,ad}q\Q43{q^{-n},abcdq^{n-1},de^{i\phi},de^{-i\phi}}{ad,bd,cd}q\\
&=\sum_{0\leq m}\frac{(q^{-n},abcdq^{n-1},d e^{i\theta},de^{-i\theta},de^{i\phi},de^{-i\phi};q)_{m}}{(q,d/a,ad,bd,cd,ad;q)_m}q^{m}\\
&\qquad\cdot\Q{10}{9}{aq^{-m}/d,q\sqrt{aq^{-m}/d},-q\sqrt{aq^{-m}/d},ae^{i\theta},ae^{-i\theta},q^{1-m}/bd,q^{-m}/cd,ae^{i\theta},ae^{-i\theta},q^{-m}}{\sqrt{aq^{-m}/d},-\sqrt{aq^{-m}/d},e^{-i\theta}q^{1-m}/d,e^{i\theta}q^{1-m}/d,ab,ac,e^{-i\phi}q^{1-m}/d,e^{i\phi}q^{1-m}/d,aq/d}{\frac{aq^2}{b_1b_2c_1c_2}}
\end{align}
ここで,
\begin{align}
&\Q43{q^{-n},abcdq^{n-1},ae^{i\theta},ae^{-i\theta}}{ab,ac,ad}q\Q43{q^{-n},abcdq^{n-1},de^{i\phi},de^{-i\phi}}{ad,bd,cd}q\\
&=\frac{(ad)^n}{(ab,ac,ad,ad,bd,cd;q)_n}p_n(x;a,b,c,d|q)p_n(y;d,a,b,c|q)
\end{align}
であること, Askey-Wilson多項式の対称性$p_n(y;d,a,b,c|q)=p_n(y;a,b,c,d|q)$を用いると定理を得る.
証明からも分かるように, terminating balanced${}_4\phi_3$の積公式 は定理1と本質的に同値である.
定理1において特に$b=a\sqrt q,d=c\sqrt q$とすると,
\begin{align}
&p_n(x;a,a\sqrt q,c,c\sqrt q|q)p_n(y;a,a\sqrt q,c,c\sqrt q|q)\\
&=(a^2\sqrt q,ac,ac\sqrt q,ac\sqrt q,acq,c^2\sqrt q;q)_n(ac\sqrt q)^{-n}\\
&\qquad\cdot\sum_{m=0}^n\frac{(q^{-n},a^2c^2q^{n},c\sqrt qe^{i\theta},c\sqrt qe^{-i\theta},c\sqrt qe^{i\phi},c\sqrt qe^{-i\phi};q)_m}{(q,ac\sqrt q,ac\sqrt q,acq,c^2\sqrt q,c\sqrt q/a;q)_m}q^m\\
&\qquad\Q{10}9{aq^{-m-\frac 12}/c,q\sqrt{aq^{-m-\frac 12}/c},-q\sqrt{aq^{-m-\frac 12}/c},q^{-m}/ac,q^{\frac 12-m}/c^2,q^{-m},ae^{i\theta},ae^{-i\theta},ae^{i\phi},ae^{-i\phi}}{\sqrt{aq^{-m-\frac 12}/c},-\sqrt{aq^{-m-\frac 12}/c},a^2\sqrt q,ac,a\sqrt q/c,e^{-i\theta}q^{\frac 12-m}/c,e^{i\theta}q^{\frac 12-m}/c,e^{-i\phi}q^{\frac 12-m}/c,e^{i\phi}q^{\frac 12-m}/c}{q}
\end{align}
となる. この右辺に現れる${}_{10}\phi_9$はvery-well-poised balancedである.
Baileyの${}_{10}\phi_9$変換公式
より, $w=a^2ce^{i\phi}/\sqrt q$として
\begin{align}
&\Q{10}9{aq^{-m-\frac 12}/c,q\sqrt{aq^{-m-\frac 12}/c},-q\sqrt{aq^{-m-\frac 12}/c},q^{-m}/ac,q^{\frac 12-m}/c^2,q^{-m},ae^{i\theta},ae^{-i\theta},ae^{i\phi},ae^{-i\phi}}{\sqrt{aq^{-m-\frac 12}/c},-\sqrt{aq^{-m-\frac 12}/c},a^2\sqrt q,ac,a\sqrt q/c,e^{-i\theta}q^{\frac 12-m}/c,e^{i\theta}q^{\frac 12-m}/c,e^{-i\phi}q^{\frac 12-m}/c,e^{i\phi}q^{\frac 12-m}/c}{q}\\
&=\frac{(aq^{\frac 12-m}/c,q^{\frac 12-m}/ac,ace^{i(\phi-\theta)}\sqrt q,ace^{i(\phi+\theta)}\sqrt q;q)_m}{(e^{-i\theta}q^{\frac 12-m}/c,e^{i\theta}q^{\frac 12-m}/c,a^2ce^{i\phi}\sqrt q,ce^{i\phi}\sqrt q;q)_m}\Q{10}9{w,\sqrt wq,-\sqrt wq, ce^{i\phi},a\sqrt qe^{i\phi},ae^{i\theta},ae^{-i\theta},ae^{i\phi},a^2c^2q^m,q^{-m}}{\sqrt w,-\sqrt w,a^2\sqrt q,ac,ac\sqrt q,ace^{i(\phi+\theta)}\sqrt q,ac\sqrt q,e^{i\phi}q^{\frac 12-m}/c,a^2ce^{i\phi}q^{\frac 12+m}}q\\
&=\frac{(c\sqrt q/a,ac\sqrt q,ace^{i(\phi-\theta)}\sqrt q,ace^{i(\phi+\theta)}\sqrt q;q)_m}{(c\sqrt qe^{i\theta},c\sqrt qe^{-i\theta},a^2ce^{i\phi}\sqrt q,ce^{i\phi}\sqrt q;q)_m}\Q{10}9{w,\sqrt wq,-\sqrt wq, ce^{i\phi},a\sqrt qe^{i\phi},ae^{i\theta},ae^{-i\theta},ae^{i\phi},a^2c^2q^m,q^{-m}}{\sqrt w,-\sqrt w,a^2\sqrt q,ac,ace^{i(\phi-\theta)}\sqrt q ,ace^{i(\phi+\theta)}\sqrt q,ac\sqrt q,e^{i\phi}q^{\frac 12-m}/c,a^2ce^{i\phi}q^{\frac 12+m}}q
\end{align}
であるから, これを代入すると以下を得る.
$x=\cos\theta,y=\cos\phi, w=a^2ce^{i\phi}/\sqrt q$とするとき,
\begin{align}
&p_n(x;a,a\sqrt q,c,c\sqrt q|q)p_n(y;a,a\sqrt q,c,c\sqrt q|q)\\
&=(a^2\sqrt q,ac,ac\sqrt q,ac\sqrt q,acq,c^2\sqrt q;q)_n(ac\sqrt q)^{-n}\\
&\qquad\cdot\sum_{m=0}^n\frac{(q^{-n},a^2c^2q^{n},c\sqrt qe^{-i\phi},ace^{i(\phi-\theta)}\sqrt q,ace^{i(\phi+\theta)}\sqrt q;q)_m}{(q,ac\sqrt q,acq,c^2\sqrt q,a^2ce^{i\phi}\sqrt q;q)_m}q^m\\
&\qquad\cdot\Q{10}9{w,\sqrt wq,-\sqrt wq, ce^{i\phi},ae^{i\phi},a\sqrt qe^{i\phi},ae^{i\theta},ae^{-i\theta},a^2c^2q^m,q^{-m}}{\sqrt w,-\sqrt w,a^2\sqrt q,ac\sqrt q,ac,ace^{i(\phi-\theta)}\sqrt q,ace^{i(\phi+\theta)}\sqrt q,e^{i\phi}q^{\frac 12-m}/c,a^2ce^{i\phi}q^{\frac 12+m}}q
\end{align}
が成り立つ.
