三次変換公式から得られる積分を7F6和公式に書き換える
今回は前の記事(
三次変換公式から従う2F1の積を含む積分
)の定理1の2つ目の等式
\begin{align}
&\int_0^1(x(1-x))^{2d-\frac 23}\F21{3d,d+\frac 13}{2d+\frac 23}
{x}\F21{3d,d+\frac 13}{2d+\frac 23}{1-x}\,dx\\
&=\frac{2^{8d+\frac 13}\pi\Gamma\left(\frac 13\right)^2\Gamma\left(3d+\frac 12\right)^2\Gamma\left(d+\frac 16\right)}{3^{9d+\frac 12}\Gamma\left(\frac23\right)\Gamma\left(d+\frac 12\right)^3\Gamma\left(2d+\frac 13\right)^2}
\end{align}
から${}_7F_6$和公式を導出したいと思う.
導出
まず, $x\mapsto 1-x$に関する対称性から,
\begin{align}
&\int_0^1(x(1-x))^{2d-\frac 23}\F21{3d,d+\frac 13}{2d+\frac 23}
{x}\F21{3d,d+\frac 13}{2d+\frac 23}{1-x}\,dx\\
&=2\int_0^{\frac 12}(x(1-x))^{2d-\frac 23}\F21{3d,d+\frac 13}{2d+\frac 23}
{x}\F21{3d,d+\frac 13}{2d+\frac 23}{1-x}\,dx
\end{align}
である. $t=4x(1-x)$とすると
\begin{align}
&\int_0^{\frac 12}(x(1-x))^{2d-\frac 23}\F21{3d,d+\frac 13}{2d+\frac 23}
{x}\F21{3d,d+\frac 13}{2d+\frac 23}{1-x}\,dx\\
&=2^{-4d-\frac 23}\int_0^1\frac{t^{2d-\frac 23}}{\sqrt{1-t}}\F21{3d,d+\frac 13}{2d+\frac 23}
{\frac{1-\sqrt{1-t}}2}\F21{3d,d+\frac 13}{2d+\frac 23}{\frac{1+\sqrt{1-t}}2}\,dt
\end{align}
となる. ここで, Clausenの公式のMellin-Barnesの積分類似(
前の記事
の定理1)より
\begin{align}
&\F21{3d,d+\frac 13}{2d+\frac 23}
{\frac{1-\sqrt{1-t}}2}\F21{3d,d+\frac 13}{2d+\frac 23}{\frac{1+\sqrt{1-t}}2}\\
&=\frac{2^{4d-\frac 23}\Gamma\left(2d+\frac 23\right)^2\cos\pi\left(d-\frac 16\right)}{\pi^{\frac 32}\Gamma(3d)\Gamma\left(d+\frac 13\right)}\\
&\qquad\cdot\frac 1{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(3d+s\right)\Gamma\left(d+\frac 13+s\right)\Gamma\left(2d+\frac 16+s\right)\Gamma\left(\frac 13-2d-s\right)\Gamma(-s)}{\Gamma\left(4d+\frac 13+s\right)}t^s\,ds
\end{align}
であるから, これを代入して積分の順序を入れ替えると,
\begin{align}
&\int_0^{\frac 12}(x(1-x))^{2d-\frac 23}\F21{3d,d+\frac 13}{2d+\frac 23}
{x}\F21{3d,d+\frac 13}{2d+\frac 23}{1-x}\,dx\\
&=\frac{2^{-\frac 43}\Gamma\left(2d+\frac 23\right)^2\cos\pi\left(d-\frac 16\right)}{\pi^{\frac 32}\Gamma(3d)\Gamma\left(d+\frac 13\right)}\\
&\qquad\cdot\frac 1{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(3d+s\right)\Gamma\left(d+\frac 13+s\right)\Gamma\left(2d+\frac 16+s\right)\Gamma\left(\frac 13-2d-s\right)\Gamma(-s)}{\Gamma\left(4d+\frac 13+s\right)}\\
&\qquad\qquad\cdot\int_0^1\frac{t^{s+2d-\frac 23}}{\sqrt{1-t}}\,dt\,ds\\
&=\frac{2^{-\frac 43}\Gamma\left(2d+\frac 23\right)^2\cos\pi\left(d-\frac 16\right)}{\pi\Gamma(3d)\Gamma\left(d+\frac 13\right)}\\
&\qquad\cdot\frac 1{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(3d+s\right)\Gamma\left(d+\frac 13+s\right)\Gamma\left(2d+\frac 16+s\right)\Gamma\left(2d+\frac 13+s\right)\Gamma\left(\frac 13-2d-s\right)\Gamma(-s)}{\Gamma\left(4d+\frac 13+s\right)\Gamma\left(2d+\frac 56+s\right)}\,ds
\end{align}
を得る. よって冒頭の積分公式を用いると,
\begin{align}
&\frac{2^{8d+\frac 13}\pi\Gamma\left(\frac 13\right)^2\Gamma\left(3d+\frac 12\right)^2\Gamma\left(d+\frac 16\right)}{3^{9d+\frac 12}\Gamma\left(\frac23\right)\Gamma\left(d+\frac 12\right)^3\Gamma\left(2d+\frac 13\right)^2}\\
&=\frac{2^{-\frac 13}\Gamma\left(2d+\frac 23\right)^2\cos\pi\left(d-\frac 16\right)}{\pi\Gamma(3d)\Gamma\left(d+\frac 13\right)}\\
&\qquad\cdot\frac 1{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(3d+s\right)\Gamma\left(d+\frac 13+s\right)\Gamma\left(2d+\frac 16+s\right)\Gamma\left(2d+\frac 13+s\right)\Gamma\left(\frac 13-2d-s\right)\Gamma(-s)}{\Gamma\left(4d+\frac 13+s\right)\Gamma\left(2d+\frac 56+s\right)}\,ds
\end{align}
を得る. ガンマ関数の相反公式, 乗法公式を用いて整理すると以下を得る.
\begin{align} &\frac 1{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(3d+s\right)\Gamma\left(d+\frac 13+s\right)\Gamma\left(2d+\frac 16+s\right)\Gamma\left(2d+\frac 13+s\right)\Gamma\left(\frac 13-2d-s\right)\Gamma(-s)}{\Gamma\left(4d+\frac 13+s\right)\Gamma\left(2d+\frac 56+s\right)}\,ds\\ &=\frac{\Gamma\left(\frac 13\right)^3\Gamma(d)\Gamma\left(d+\frac 16\right)}{2^{\frac 43}\sqrt 3\Gamma\left(d+\frac 12\right)\Gamma\left(d+\frac 23\right)\cos\pi\left(d-\frac 16\right)} \end{align}
Non-terminating Whippleの変換公式
を用いると定理1の左辺は以下のように書き換えられる.
\begin{align}
&\frac 1{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma\left(3d+s\right)\Gamma\left(d+\frac 13+s\right)\Gamma\left(2d+\frac 16+s\right)\Gamma\left(2d+\frac 13+s\right)\Gamma\left(\frac 13-2d-s\right)\Gamma(-s)}{\Gamma\left(4d+\frac 13+s\right)\Gamma\left(2d+\frac 56+s\right)}\,ds\\
&=\frac{\Gamma\left(d+\frac 13\right)\Gamma\left(2d+\frac 16\right)\Gamma\left(2d+\frac 13\right)\Gamma(3d)\Gamma\left(\frac 23\right)\Gamma\left(\frac 12\right)\Gamma\left(\frac 23-d\right)\Gamma\left(3d+\frac 76\right)}{\Gamma\left(4d+\frac 13\right)\Gamma\left(2d+\frac 56\right)^2\Gamma(d+1)\Gamma\left(d+\frac 56\right)}\\
&\qquad\cdot\F76{3d+\frac 16,\frac{3d}2+\frac{13}{12},\frac 56-d,d+\frac 13,d+\frac 13,2d+\frac 16,2d+\frac 13}{\frac{3d}2+\frac{1}{12},4d+\frac 13,2d+\frac 56,2d+\frac 56,d+1,d+\frac 56}1
\end{align}
よってこれを定理1に代入して整理すると以下を得る.
\begin{align} &\F76{3d+\frac 16,\frac{3d}2+\frac{13}{12},\frac 56-d,d+\frac 13,d+\frac 13,2d+\frac 16,2d+\frac 13}{\frac{3d}2+\frac{1}{12},4d+\frac 13,2d+\frac 56,2d+\frac 56,d+1,d+\frac 56}1\\ &=\frac{2^{10d+\frac 13}\sqrt{\pi}\Gamma\left(\frac 13\right)^2\Gamma\left(3d+\frac 12\right)^2\Gamma\left(d+\frac 16\right)\Gamma\left(2d+\frac 56\right)^2\Gamma(d+1)}{3^{9d+\frac 12}\Gamma\left(\frac 23\right)^2\Gamma\left(d+\frac 12\right)^3\Gamma\left(2d+\frac 13\right)^3\Gamma\left(3d+\frac 76\right)} \end{align}
別の表示
まず, 定理2に ${}_7F_6$の二項変換公式 を適用すると以下の11個の別の表示が得られる.
\begin{align} &\F76{5d-\frac 16,\frac{5d}2+\frac{11}{12},d+\frac 12,3d,3d,2d+\frac 16,2d+\frac 13}{\frac{5d}2-\frac 1{12},4d+\frac 13,2d+\frac 56,2d+\frac 56,3d+\frac 23,3d+\frac 12}1\\ &=\frac{2^{4d-\frac 23}\Gamma\left(\frac 13\right)^2\Gamma\left(\frac 23-d\right)\Gamma\left(d+\frac 16\right)\Gamma\left(d+\frac 56\right)^2\Gamma\left(2d+\frac 56\right)^2\Gamma\left(3d+\frac 23\right)}{\sqrt{3}\pi\Gamma\left(\frac 23\right)^2 \Gamma\left(d+\frac 13\right)\Gamma\left(d+\frac 23\right)^3\Gamma\left(5d+\frac 56\right)}\\ &\F76{5d-\frac 13,\frac{5d}2+\frac 56,d+\frac 13,3d,3d,2d+\frac 16,2d}{\frac{5d}2-\frac 16,4d+\frac 13,2d+\frac 23,2d+\frac 23,3d+\frac 12,3d+\frac 23}1\\ &=\frac{2^{12d}\Gamma\left(\frac 13\right)^2\Gamma\left(3d+\frac 12\right)^3\Gamma\left(2d+\frac 23\right)\Gamma\left(d+\frac 16\right)\Gamma\left(\frac 56-d\right)\Gamma\left(3d+\frac 23\right)}{3^{9d+\frac 12}\Gamma\left(\frac 23\right)^2\Gamma\left(d+\frac 12\right)^3\Gamma\left(2d+\frac 13\right)^3\Gamma\left(5d+\frac 23\right)}\\ &\F76{4d,2d+1,\frac 23,2d+\frac 16,3d,d+\frac 13,2d+\frac 13}{2d,4d+\frac 13,2d+\frac 56,d+1,3d+\frac 23,2d+\frac 23}1\\ &=\frac{2^{6d}\Gamma\left(\frac 13\right)^2\Gamma(d+1)\Gamma\left(d+\frac 56\right)^2\Gamma\left(2d+\frac 56\right)\Gamma\left(3d+\frac 23\right)}{3^{3d+\frac 12}\Gamma\left(\frac 23\right)^2\Gamma\left(d+\frac 12\right)\Gamma\left(d+\frac 23\right)^3\Gamma(4d+1)}\\ &\F76{4d-\frac 16,2d+\frac{11}{12},\frac 12,2d,3d,d+\frac 13,2d+\frac 16}{2d-\frac 1{12},4d+\frac 13,2d+\frac 56,d+\frac 56,3d+\frac 12,2d+\frac 23}1\\ &=\frac{2^{6d-2}\Gamma\left(\frac 13\right)^3\Gamma\left(d+\frac 16\right)\Gamma\left(d+\frac 56\right)^4\Gamma\left(2d+\frac 56\right)}{\pi^{\frac 52}\Gamma\left(d+\frac 23\right)^3\Gamma\left(4d+\frac 56\right)}\\ &\F76{3d,1+\frac{3d}2,\frac 23-d,2d,2d+\frac 16,d+\frac 13,d+\frac 13}{\frac{3d}2,4d+\frac 13,d+1,d+\frac 56,2d+\frac 23,2d+\frac 23}1\\ &=\frac{2^{8d+\frac 23}\Gamma\left(\frac 13\right)^2\Gamma\left(d+\frac 56\right)^4}{3^{6d+1}\Gamma\left(\frac 23\right)^2\Gamma\left(d+\frac 23\right)^4}\\ &\F76{3d+\frac 16,\frac{3d}2+\frac{13}{12},d+\frac 13,3d,d+\frac 12,\frac 23,\frac 12}{\frac{3d}2+\frac 1{12},2d+\frac 56,\frac 76,2d+\frac 23,3d+\frac 12,3d+\frac 23}1\\ &=\frac{2^{6d+\frac 23}\pi^{\frac 32}\Gamma\left(\frac 13\right)^3\Gamma\left(3d+\frac 12\right)^3\Gamma\left(d+\frac 16\right)\Gamma\left(2d+\frac 56\right)\Gamma\left(3d+\frac 23\right)}{3^{9d+\frac 32}\Gamma\left(\frac 23\right)^3\Gamma\left(d+\frac 12\right)^3\Gamma\left(2d+\frac 13\right)^3\Gamma\left(3d+\frac 76\right)\Gamma\left(2d+\frac 16\right)\Gamma\left(d+\frac 56\right)}\\ &\F76{2d+\frac 12,d+\frac 54,\frac 23,\frac 23,d+\frac 12,\frac 56-d,2d+\frac 13}{d+\frac 14,2d+\frac 56,2d+\frac 56,d+1,3d+\frac 23,\frac 76}1\\ &=\frac{2^{6d-1}\Gamma\left(\frac 13\right)^3\Gamma(d)\Gamma(d+1)\Gamma\left(d+\frac 56\right)\Gamma\left(2d+\frac 56\right)^2\Gamma\left(3d+\frac 23\right)}{3^{3d+\frac 32}\Gamma\left(\frac 23\right)^3\Gamma\left(d+\frac 13\right)\Gamma\left(d+\frac 23\right)^3\Gamma\left(2d+\frac 32\right)\Gamma\left(4d+\frac 13\right)}\\ &\F76{2d+\frac 13,d+\frac 76,\frac 12,\frac 12,d+\frac 12,\frac 56-d,2d+\frac 16}{d+\frac 16,2d+\frac 56,2d+\frac 56,d+\frac 56,3d+\frac 12,\frac 76}1\\ &=\frac{2^{-2d}\sqrt{\pi}\Gamma\left(\frac 13\right)^3\Gamma\left(d+\frac 16\right)\Gamma\left(d+\frac 56\right)^2\Gamma\left(2d+\frac 56\right)^2}{3^{\frac 32}\Gamma\left(\frac 23\right)^3\Gamma\left(d+\frac 13\right)^2\Gamma\left(d+\frac 23\right)^3\Gamma\left(2d+\frac 43\right)}\\ &\F76{2d+\frac 13,d+\frac 76,d+\frac 13,2d+\frac 16,\frac 23,\frac 23,\frac 23-d}{d+\frac 16,d+1,\frac 76,2d+\frac 23,2d+\frac 23,3d+\frac 23}1\\ &=\frac{2^{8d+\frac 13}\pi\Gamma\left(\frac 13\right)^3\Gamma\left(d+\frac 16\right)\Gamma(d+1)\Gamma\left(3d+\frac 12\right)^2\Gamma\left(3d+\frac 23\right)}{3^{9d+\frac 32}\Gamma\left(\frac 23\right)^3\Gamma\left(d+\frac 12\right)^3\Gamma\left(2d+\frac 13\right)^3\Gamma\left(2d+\frac 43\right)}\\ &\F76{2d+\frac 16,d+\frac{13}{12},d+\frac 13,2d,\frac 12,\frac 12,\frac 23-d}{d+\frac 1{12},d+\frac 56,\frac 76,2d+\frac 23,2d+\frac 23,3d+\frac 12}1\\ &=\frac{2^{8d-\frac 83}\Gamma\left(\frac 13\right)^3\Gamma\left(d+\frac 16\right)^4\Gamma\left(d+\frac 56\right)^4}{3^{\frac 32}\pi^2\Gamma\left(\frac 23\right)^3\Gamma\left(2d+\frac 13\right)^2\Gamma\left(2d+\frac 16\right)\Gamma\left(2d+\frac 76\right)}\\ &\F76{d+\frac 12,\frac d2+\frac 54,\frac 23-d,\frac 12,\frac 23,\frac 56-d,d+\frac 13}{\frac d2+\frac 14,2d+\frac 56,d+1,d+\frac 56,2d+\frac 23,\frac 76}1\\ &=\frac{\Gamma\left(\frac 13\right)^3\Gamma(d)\Gamma(d+1)\Gamma\left(d+\frac 56\right)^2\Gamma\left(2d+\frac 56\right)}{9\cdot 2^{\frac 13}\Gamma\left(\frac 23\right)^3\Gamma\left(d+\frac 12\right)\Gamma\left(d+\frac 32\right)\Gamma\left(d+\frac 23\right)^2\Gamma\left(2d+\frac 16\right)} \end{align}
書いていて分量が多くなりそうだと思ったので, 以下
\begin{align}
W(a;b,c,d,e,f):=\F76{a,1+\frac a2,b,c,d,e,f}{\frac a2,1+a-b,1+a-c,1+a-d,1+a-e,1+a-f}1
\end{align}
という略記を用いる. Baileyの三項変換公式(
前の記事
の定理2)において, $a,b,c,d,e,f$を$3d+\frac 16,\frac 23,\frac 12,3d,2d+\frac 16,2d+\frac 13$と置き換えると
\begin{align}
&W\left(3d+\frac 16;\frac 23,\frac 12,3d,2d+\frac 16,2d+\frac 13\right)
\end{align}
を
\begin{align}
&W\left(4d;3d,\frac 23,d+\frac 13,2d+\frac 16,2d+\frac 13\right),\\
&W\left(\frac 76-3d;\frac 23,1-3d,\frac 12,\frac 23-d,\frac 56-d\right)
\end{align}
で表す式を得る. 1つ目は定理3の3つ目の式から, 2つ目は定理3の6つ目の式において$d\mapsto \frac 13-d$としたものからガンマ積で表される. 整理すると
\begin{align}
&W\left(3d+\frac 16;\frac 23,\frac 12,3d,2d+\frac 16,2d+\frac 13\right)\\
&=\frac{\Gamma\left(\frac 13\right)^4\Gamma\left(d+\frac 12\right)\Gamma\left(d+\frac 56\right)\Gamma(d+1)\Gamma\left(2d+\frac 23\right)\Gamma\left(\frac 43-2d\right)\Gamma\left(3d+\frac 23\right)}{6\Gamma\left(\frac 23\right)^2\Gamma(1-d)\Gamma\left(\frac 76-d\right)\Gamma\left(d+\frac 13\right)^2\Gamma\left(d+\frac 23\right)^3\Gamma\left(3d+\frac 76\right)}
\end{align}
を得る. それらに
${}_7F_6$の二項変換公式
を適用して得られる和公式は以下のようになる.
\begin{align} W\left(3d+\frac 16;\frac 23,\frac 12,3d,2d+\frac 16,2d+\frac 13\right)&=\frac{\Gamma\left(\frac 13\right)^4\Gamma\left(d+\frac 12\right)\Gamma\left(d+\frac 56\right)\Gamma(d+1)\Gamma\left(2d+\frac 23\right)\Gamma\left(\frac 43-2d\right)\Gamma\left(3d+\frac 23\right)}{6\Gamma\left(\frac 23\right)^2\Gamma(1-d)\Gamma\left(\frac 76-d\right)\Gamma\left(d+\frac 13\right)^2\Gamma\left(d+\frac 23\right)^3\Gamma\left(3d+\frac 76\right)}\\ W\left(4d-\frac16;3d,3d,d+\frac16,d+\frac13,2d+\frac16\right) &= \frac{ 2^{2d-1}\Gamma\left(\frac13\right)^3 \Gamma\left(d+\frac12\right) \Gamma\left(d+\frac56\right)^4 \Gamma(1-2d) \Gamma\left(3d+\frac23\right)} { \pi^2\Gamma\left(\frac23\right) \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 \Gamma\left(4d+\frac56\right) } \\ W\left(2d+\frac13;\frac23,3d,\frac23-d,d+\frac13,d+\frac12\right) &= \frac{ \Gamma\left(\frac13\right)^3 \Gamma(2d+1) \Gamma\left(d+\frac56\right)^3 \Gamma\left(\frac43-d\right) \Gamma\left(3d+\frac23\right)} { 2\pi\Gamma\left(\frac23\right) \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 \Gamma\left(2d+\frac43\right) \Gamma\left(3d+\frac12\right) } \\ W\left(d+\frac12;\frac23,\frac23,\frac56-2d,d+\frac13,2d+\frac16\right) &= \frac{ \Gamma\left(\frac13\right)^4 \Gamma\left(d+\frac56\right)^3 \Gamma\left(\frac43-d\right) \Gamma\left(3d+\frac23\right)} { 3\pi(2d+1)\Gamma\left(\frac23\right)^2 \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 \Gamma\left(3d+\frac12\right) } \\ W\left(2d+\frac16;\frac12,3d,\frac23-d,d+\frac16,d+\frac13\right) &= \frac{ \Gamma\left(\frac13\right)^3 \Gamma(2d+1) \Gamma\left(d+\frac56\right)^3 \Gamma\left(\frac76-d\right)} { 2\pi^{\frac 32} \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 \Gamma\left(2d+\frac76\right) } \\ W\left(d+\frac13;\frac12,\frac12,\frac56-2d,d+\frac16,2d+\frac16\right) &= \frac{ \Gamma\left(\frac13\right)^4 \Gamma\left(d+\frac12\right)^2 \Gamma\left(d+\frac56\right)^3 \Gamma\left(\frac76-d\right)} { 6\pi\Gamma\left(\frac23\right)^2 \Gamma(1-d) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^3 \Gamma\left(d+\frac43\right) } \\ W\left(\frac56-d;\frac12,\frac23,\frac23-d,\frac56-2d,1-2d\right) &= \frac{ \Gamma\left(\frac13\right)^4 \Gamma(2d+1) \Gamma\left(d+\frac56\right)^2 \Gamma\left(\frac76-d\right) \Gamma(3d) \Gamma\left(\frac43-d\right)} { 6\cdot 2^{2d}\sqrt{\pi}\, \Gamma\left(\frac23\right)^2 \Gamma(1-d) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^3 \Gamma\left(\frac{11}{6}-d\right) \Gamma\left(3d+\frac12\right) } \\ W\left(4d;3d,3d,d+\frac13,d+\frac12,2d+\frac13\right) &= \frac{ \Gamma\left(\frac13\right)^3 \Gamma(2d+1) \Gamma(d+1) \Gamma\left(d+\frac56\right)^2 \Gamma\left(\frac56-2d\right) \Gamma\left(3d+\frac23\right)} { 2\pi^{\frac 32}\Gamma\left(\frac23\right) \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 \Gamma(4d+1) } \\ W\left(d+\frac23;\frac23,\frac23,1-2d,d+\frac12,2d+\frac13\right) &= \frac{ \Gamma\left(\frac13\right)^4 \Gamma(2d+1) \Gamma(d+1) \Gamma\left(d+\frac16\right) \Gamma\left(d+\frac56\right) \Gamma\left(\frac43-d\right) \Gamma\left(3d+\frac23\right)} { 6\cdot 2^{2d}\sqrt{\pi}\, \Gamma\left(\frac23\right)^2 \Gamma(1-d) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^3 \Gamma\left(d+\frac53\right) \Gamma\left(3d+\frac12\right) } \\ W\left(d+\frac12;\frac12,\frac12,1-2d,d+\frac13,2d+\frac13\right) &= \frac{ \Gamma\left(\frac13\right)^4 \Gamma(d+1)^2 \Gamma\left(d+\frac56\right) \Gamma\left(\frac76-d\right)} { 3\pi(2d+1)\Gamma\left(\frac23\right)^2 \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 } \\ W\left(\frac12;\frac23-d,\frac23-d,\frac56-2d,d+\frac16,d+\frac13\right) &= \frac{ 2^{2d}\Gamma\left(\frac13\right)^3 \Gamma\left(d+\frac12\right) \Gamma\left(d+\frac56\right)^4 \Gamma\left(\frac76-d\right) \Gamma\left(\frac43-d\right) \Gamma\left(2d+\frac13\right)} { \pi^{\frac 52}\Gamma\left(\frac23\right) \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 \Gamma\left(3d+\frac12\right) } \\ W\left(\frac23;\frac23-d,\frac23-d,1-2d,d+\frac13,d+\frac12\right) &= \frac{ 3\Gamma\left(\frac13\right)^3 \Gamma(2d+1) \Gamma(d+1) \Gamma\left(d+\frac56\right)^2 \Gamma\left(\frac76-d\right) \Gamma\left(\frac43-d\right) \Gamma\left(2d+\frac16\right)} { 4\pi^{\frac 32}\Gamma\left(\frac23\right)^2 \Gamma(1-d) \Gamma\left(d+\frac23\right)^3 \Gamma\left(3d+\frac12\right) } \end{align}
次に, Baileyの三項変換公式(
前の記事
の定理2)において, $a,b,c,d,e,f$を$d+\frac 12,\frac 56-d,\frac 23-d,2d+\frac 16,2d+\frac 13,d+\frac 13$と置き換えると
\begin{align}
W\left(d+\frac 12;\frac 56-d,\frac 23-d,2d+\frac 16,2d+\frac 13,d+\frac 13\right)
\end{align}
を
\begin{align}
&W\left(4d;3d,\frac 23,d+\frac 13,2d+\frac 16,2d+\frac 13\right),\\
&W\left(\frac 76-3d;\frac 23,1-3d,\frac 12,\frac 23-d,\frac 56-d\right)
\end{align}
で表す式を得る. これは先ほど定理4を示す際に用いたもので, 同じようにガンマ積で評価して和公式が得られる. それらに
${}_7F_6$の二項変換公式
を適用して得られる和公式は以下のようになる.
\begin{align} W\left(d+\frac12;\frac23-d,\frac56-d,d+\frac13,2d+\frac16,2d+\frac13\right) &= \frac{\Gamma\left(\frac13\right)^4 \Gamma\left(2d+\frac23\right) \Gamma\left(2d+\frac56\right) \Gamma\left(\frac76-d\right) \Gamma\left(\frac43-d\right)} {6\sqrt{\pi}\, \Gamma\left(\frac23\right) \Gamma(1-d)^2 \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2 \Gamma\left(d+\frac32\right)}\\ W\left(\frac76-3d;1-3d,\frac56-2d,1-2d,\frac23-d,\frac56-d\right) &= \frac{\Gamma\left(\frac13\right)^4 \Gamma\left(\frac43-2d\right) \Gamma\left(\frac32-2d\right) \Gamma\left(\frac76-d\right) \Gamma\left(\frac43-d\right) \Gamma(3d)} {6\sqrt{\pi}\, \Gamma\left(\frac{13}{6}-3d\right) \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(\frac23\right) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2}\\ W\left(\frac23-d;1-3d,\frac23-d,\frac13,\frac12,d+\frac13\right) &= \frac{\Gamma\left(\frac13\right)^3 \Gamma\left(\frac43-2d\right) \Gamma\left(\frac76-d\right) \Gamma\left(\frac43-d\right) \Gamma\left(d+\frac12\right) \Gamma\left(2d+\frac23\right)} {2^{\frac{2}{3}}\pi\, \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(\frac53-d\right) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2}\\ W\left(\frac12;\frac56-2d,\frac23-d,\frac13,d+\frac13,2d+\frac16\right) &= \frac{\Gamma\left(\frac13\right)^4 \Gamma\left(\frac43-2d\right) \Gamma\left(\frac76-d\right) \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac23\right)} {3\pi\, \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2}\\ W\left(\frac23;1-2d,\frac23-d,\frac12,d+\frac13,2d+\frac13\right) &= \frac{\Gamma\left(\frac13\right)^4 \Gamma\left(\frac43-2d\right) \Gamma\left(\frac43-d\right) \Gamma(d+1) \Gamma\left(2d+\frac23\right)} {4\Gamma\left(\frac23\right)^2 \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2}\\ W\left(\frac56-d;1-3d,\frac56-d,\frac12,\frac23,d+\frac13\right) &= \frac{\Gamma\left(\frac13\right)^3 \Gamma\left(\frac32-2d\right) \Gamma\left(\frac76-d\right) \Gamma\left(\frac43-d\right) \Gamma\left(2d+\frac56\right)} {2^{\frac{2}{3}}\pi\, \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(\frac{11}{6}-d\right) \Gamma\left(d+\frac23\right)^2}\\ W\left(\frac23;\frac56-2d,\frac56-d,\frac12,d+\frac12,2d+\frac16\right) &= \frac{\Gamma\left(\frac13\right)^4 \Gamma\left(\frac32-2d\right) \Gamma\left(\frac76-d\right) \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac56\right)} {4\Gamma\left(\frac23\right)^2 \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2}\\ W\left(\frac56;1-2d,\frac56-d,\frac23,d+\frac12,2d+\frac13\right) &= \frac{\Gamma\left(\frac13\right)^6 \Gamma\left(\frac32-2d\right) \Gamma\left(\frac43-d\right) \Gamma(d+1) \Gamma\left(2d+\frac56\right)} {5\cdot2^{\frac{1}{3}}\pi\, \Gamma\left(\frac23\right)^2 \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2}\\ W\left(2d+\frac16;\frac13,\frac12,d+\frac13,2d+\frac16,3d\right) &= \frac{\Gamma\left(\frac13\right)^3 \Gamma(1-2d) \Gamma\left(\frac76-d\right) \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac23\right) \Gamma\left(2d+\frac56\right)} {2^{\frac{2}{3}}\pi\, \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2 \Gamma\left(2d+\frac76\right)}\\ W\left(2d+\frac13;\frac12,\frac23,d+\frac13,2d+\frac13,3d\right) &= \frac{\Gamma\left(\frac13\right)^3 \Gamma\left(\frac56-2d\right) \Gamma\left(\frac43-d\right) \Gamma(d+1) \Gamma\left(2d+\frac23\right) \Gamma\left(2d+\frac56\right)} {2^{\frac{2}{3}}\pi\, \Gamma\left(\frac23-d\right) \Gamma(1-d)^2 \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right)^2 \Gamma\left(2d+\frac43\right)}\\ \end{align}
同様に三項変換公式, 二項変換公式を適用して得られる和公式を以下にまとめておく. 各定理は二項変換公式を適用して移り合うものをまとめている.
\begin{align} W\left(\frac12-d;d+\frac13,d+\frac16,\frac23-d,\frac56-2d,\frac23-2d\right) &= \frac{ 2^{\frac53-4d} \Gamma\left(\frac13\right) \Gamma\left(\frac76-d\right)^2 \Gamma\left(\frac76-2d\right) \Gamma\left(d+\frac56\right) }{ \sqrt{3\pi} \Gamma\left(d+\frac13\right) \Gamma(1-d) \Gamma\left(\frac32-d\right) \Gamma\left(\frac43-2d\right) } \\ W\left(\frac56-3d;1-3d,\frac23-2d,\frac56-2d,\frac12-d,\frac23-d\right) &= \frac{ 2^{\frac43-2d} \sin\pi\left(d+\frac13\right) \Gamma\left(\frac13\right) \Gamma(3d) \Gamma\left(\frac76-2d\right) \Gamma\left(\frac76-d\right)^2 }{ \sqrt3\pi \Gamma\left(\frac{11}{6}-3d\right) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right) } \\ W\left(\frac23-2d;1-3d,\frac23-2d,\frac23-d,\frac13,\frac12\right) &= \frac{ 2^{\frac43-2d} \sin\pi\left(d+\frac13\right) \Gamma\left(\frac13\right) \Gamma\left(\frac76-2d\right) \Gamma\left(\frac76-d\right) \Gamma\left(2d+\frac16\right) }{ \sqrt3\pi \Gamma\left(\frac56\right) \Gamma\left(\frac53-2d\right) \Gamma\left(d+\frac13\right) } \\ W\left(\frac56-2d;1-3d,\frac56-2d,\frac23-d,\frac12,\frac23\right) &= \frac{ 2^{\frac13-2d}(1-6d) \tan\pi\left(d+\frac13\right) \Gamma\left(\frac13\right) \Gamma(2d) \Gamma\left(\frac76-2d\right) \Gamma\left(\frac76-d\right) }{ 3\sqrt3 \Gamma\left(\frac56\right) \Gamma(1-d) \Gamma\left(\frac{11}{6}-2d\right) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac23\right) } \\ W\left(\frac16;\frac23-2d,\frac12-d,\frac13,d+\frac16,2d\right) &= \frac{ 8\sin^2\pi\left(d+\frac13\right) \Gamma\left(\frac76-2d\right) \Gamma\left(2d+\frac12\right) }{ \sqrt\pi \Gamma\left(\frac16\right) } \\ W\left(\frac13;\frac56-2d,\frac12-d,\frac12,d+\frac16,2d+\frac16\right) &= \frac{ (1-6d) \sin^2\pi\left(d+\frac13\right) \Gamma\left(\frac76-2d\right) \Gamma\left(2d+\frac12\right) }{ \sqrt3 \cos\pi\left(d+\frac13\right) \Gamma(1-d) \Gamma\left(d+\frac23\right) } \\ W\left(d+\frac16;\frac23-d,\frac13,\frac12,d+\frac16,3d\right) &= \frac{ 2^{\frac43-2d}\pi(6d-1) \Gamma\left(\frac13\right) \Gamma\left(\frac76-2d\right) \Gamma\left(2d+\frac12\right) }{ 3\sqrt3 \cos\pi\left(d+\frac13\right) \Gamma\left(\frac56\right) \Gamma(1-d) \Gamma\left(\frac73-2d\right) \Gamma\left(d-\frac23\right) \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac76\right) } \\ W\left(\frac13;\frac23-2d,\frac23-d,\frac12,d+\frac13,2d\right) &= \frac{1-6d}{\sqrt3} \tan\pi\left(d+\frac13\right) \\ W\left(\frac12;\frac56-2d,\frac23-d,\frac23,d+\frac13,2d+\frac16\right) &= \frac{ (6d-1)^2 \sin\pi\left(d+\frac13\right) \Gamma\left(\frac13\right)^2 }{ 9\sqrt3 \cos^2\pi\left(d+\frac13\right) \Gamma(1-d) \Gamma\left(d+\frac23\right) } \\ W\left(d+\frac13;\frac23-d,\frac12,\frac23,d+\frac13,3d\right) &= \frac{ (1-6d) \tan\pi\left(d+\frac13\right) \Gamma\left(\frac13\right) \Gamma\left(\frac12-d\right) \Gamma\left(d+\frac56\right) }{ 3\sqrt{3\pi} \Gamma\left(\frac56\right) \Gamma(1-d) \Gamma\left(d+\frac43\right) } \end{align}
$C:=5\sin\pi d+\sqrt 3\cos\pi d$とする.
\begin{align}
W\left(3d-\frac16;\frac12,\frac13,2d,2d+\frac16,3d\right)
&=
\frac{
\Gamma\left(\frac23-d\right)
\Gamma\left(d+\frac16\right)
\Gamma\left(d+\frac56\right)^2
\Gamma\left(3d+\frac13\right)
}{
2\sqrt3\pi
\Gamma\left(d+\frac13\right)
\Gamma\left(d+\frac23\right)
\Gamma\left(3d+\frac56\right)
}
C
\\
W\left(4d-\frac16;d+\frac13,d+\frac12,2d+\frac16,3d,3d\right)
&=
\frac{
2^{2d-\frac43}
\Gamma\left(\frac23-2d\right)
\Gamma\left(d+\frac16\right)
\Gamma\left(d+\frac56\right)^4
\Gamma\left(3d+\frac13\right)
}{
\sqrt3\pi^{\frac32}
\Gamma\left(\frac56\right)
\Gamma\left(d+\frac23\right)^2
\Gamma\left(4d+\frac56\right)
}
C
\\
W\left(4d-\frac13;d+\frac16,d+\frac13,2d,3d,3d\right)
&=
\frac{
2^{2d-\frac43}
\Gamma\left(\frac56-2d\right)
\Gamma\left(d+\frac16\right)
\Gamma\left(d+\frac56\right)^2
\Gamma\left(3d+\frac13\right)
}{
\sqrt3\pi^{\frac32}
\Gamma\left(\frac56\right)
\Gamma\left(4d+\frac23\right)
}
C
\\
W\left(2d;\frac23-d,\frac13,d+\frac16,d+\frac13,3d\right)
&=
\frac{
2^{2d-\frac43}
\Gamma(1-d)
\Gamma\left(d+\frac16\right)
\Gamma\left(d+\frac56\right)^3
\Gamma\left(3d+\frac13\right)
}{
\sqrt3\pi
\Gamma\left(\frac56\right)
\Gamma\left(d+\frac23\right)
\Gamma(2d+1)
\Gamma\left(3d+\frac12\right)
}
C
\\
W\left(d+\frac16;\frac56-2d,\frac13,\frac13,d+\frac13,2d+\frac16\right)
&=
\frac{
\sqrt3
\Gamma(1-d)
\Gamma\left(d+\frac56\right)^3
\Gamma\left(3d+\frac13\right)
}{
2\pi
\Gamma\left(d+\frac23\right)^2
\Gamma\left(3d+\frac32\right)
}
C
\\
W\left(d;\frac23-2d,\frac13,\frac13,d+\frac16,2d\right)
&=
\frac{
\Gamma(1-d)
\Gamma\left(3d+\frac13\right)
}{
3^{3d+\frac12}
\Gamma\left(d+\frac13\right)
\Gamma(d+1)
}
C
\\
W\left(2d+\frac16;\frac23-d,\frac12,d+\frac13,d+\frac12,3d\right)
&=
\frac{
2^{2d-\frac73}(1-6d)
\Gamma\left(\frac13\right)
\Gamma\left(d+\frac16\right)
\Gamma\left(d+\frac56\right)^2
}{
3\sqrt{3\pi}
\cos\pi\left(d+\frac13\right)
\Gamma\left(\frac56\right)
\Gamma\left(d+\frac23\right)
\Gamma\left(2d+\frac76\right)
}
C
\\
W\left(d+\frac13;\frac56-2d,\frac12,\frac12,d+\frac12,2d+\frac16\right)
&=
\frac{
(1-6d)
\Gamma\left(d+\frac16\right)^2
\Gamma\left(d+\frac56\right)^2
}{
12\sqrt3
\cos\pi\left(d+\frac13\right)
\Gamma\left(d+\frac13\right)
\Gamma\left(d+\frac23\right)^2
\Gamma\left(d+\frac43\right)
}
C
\\
W\left(d+\frac16;\frac23-2d,\frac12,\frac12,d+\frac13,2d\right)
&=
\frac{
1-6d
}{
2\sqrt3(6d+1)
\cos\pi\left(d+\frac13\right)
}
C
\\
W\left(\frac12-d;\frac23-2d,\frac56-2d,\frac23-d,\frac13,\frac12\right)
&=
-\frac{
\Gamma(3d)
\Gamma(1-d)
\Gamma\left(d+\frac16\right)
\Gamma\left(d+\frac56\right)
}{
6\sqrt3
\cos\pi\left(d+\frac13\right)
\Gamma\left(\frac32-d\right)
\Gamma\left(d+\frac13\right)
\Gamma\left(d+\frac23\right)
\Gamma\left(3d-\frac12\right)
}
C
\\
W\left(\frac12;\frac56-2d,\frac23-d,\frac23-d,d+\frac13,d+\frac12\right)
&=
\frac{
2^{\frac13-2d}
\Gamma\left(\frac56\right)
\Gamma\left(\frac23-d\right)^2
\Gamma\left(d+\frac16\right)
\Gamma\left(3d+\frac13\right)^2
\Gamma\left(3d+\frac12\right)
}{
4\sqrt3
\Gamma(2d)
\Gamma(1-d)
\Gamma\left(\frac76-d\right)
\Gamma\left(d+\frac13\right)^2
\Gamma\left(3d+\frac56\right)^2
}
C
\\
W\left(\frac13;\frac23-2d,\frac23-d,\frac23-d,d+\frac16,d+\frac13\right)
&=
\frac{
2^{\frac23-2d}
\Gamma\left(\frac23\right)
\Gamma\left(\frac23-d\right)^2
\Gamma\left(d+\frac16\right)
\Gamma\left(d+\frac56\right)^2
\Gamma\left(3d+\frac13\right)^2
\Gamma\left(3d+\frac12\right)
}{
6\sqrt3
\Gamma(1-d)
\Gamma\left(\frac76-d\right)
\Gamma\left(d+\frac13\right)^2
\Gamma\left(d+\frac23\right)^2
\Gamma\left(2d+\frac16\right)
\Gamma\left(3d+\frac56\right)^2
}
C
\end{align}
\begin{align} W\left(3d-\frac16;\frac12-d,2d,2d+\frac16,d+\frac13,d+\frac13\right) &= \frac{ 2^{10d+\frac13} \Gamma\left(\frac13\right)^2 \Gamma\left(d+\frac23\right) \Gamma\left(2d+\frac12\right)^2 \Gamma\left(3d+\frac12\right)^2 }{ 3^{9d}\sqrt{\pi} \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac13\right)^3 \Gamma\left(3d+\frac56\right) } \\ W\left(4d-\frac13;\frac13,d+\frac13,2d,2d+\frac16,3d\right) &= \frac{ 2^{10d-\frac13} \Gamma\left(\frac13\right)^2 \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac12\right) \Gamma\left(3d+\frac13\right) \Gamma\left(3d+\frac12\right) }{ 3^{6d}\pi \Gamma\left(d+\frac12\right) \Gamma\left(2d+\frac13\right)^2 \Gamma\left(4d+\frac23\right) } \\ W\left(4d-\frac16;\frac12,d+\frac13,2d+\frac16,2d+\frac13,3d\right) &= \frac{ 2^{12d} \Gamma\left(\frac13\right)^3 \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac12\right) \Gamma\left(3d+\frac12\right)^3 }{ 3^{9d}\pi \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac13\right)^3 \Gamma\left(4d+\frac56\right) } \\ W\left(5d-\frac13;d+\frac13,2d+\frac16,2d+\frac13,3d,3d\right) &= \frac{ 2^{8d+1} \cos(\pi d) \Gamma\left(\frac13\right)^2 \Gamma\left(2d+\frac12\right)^4 \Gamma\left(5d+\frac23\right) }{ 3^{6d}\pi \Gamma\left(d+\frac16\right) \Gamma\left(d+\frac13\right) \Gamma\left(2d+\frac13\right) \Gamma\left(3d+\frac56\right) \Gamma\left(6d+\frac23\right) } \\ W\left(d+\frac16;\frac13,\frac12,\frac12-d,\frac23-d,d+\frac13\right) &= \frac{ 2^{4d+3}\sqrt{\pi} \Gamma(2d) \Gamma\left(2d+\frac23\right) \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac12\right) \Gamma\left(3d+\frac12\right) }{ 3^{3d+1} \Gamma\left(d+\frac16\right)^2 \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac43\right) \Gamma\left(4d+\frac13\right) } \\ W\left(2d-\frac16;\frac12-d,\frac13,\frac13,d+\frac16,2d\right) &= \frac{ 2^{14d+\frac53}\sqrt{\pi} \Gamma\left(2d+\frac12\right)^2 \Gamma\left(3d+\frac13\right) \Gamma\left(3d+\frac12\right)^2 }{ 3^{9d+\frac12} \Gamma\left(d+\frac16\right) \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac23\right) \Gamma\left(4d+\frac13\right) \Gamma\left(4d+\frac23\right) } \\ W\left(2d;\frac12-d,\frac12,\frac12,d+\frac16,2d+\frac16\right) &= \frac{ 2^{\frac43}\pi \Gamma\left(2d+\frac12\right)^2 \Gamma\left(3d+\frac12\right)^2 }{ 3^{3d} \Gamma\left(d+\frac13\right) \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac13\right)^2 \Gamma(3d+1) } \\ W\left(2d;\frac23-d,\frac13,\frac13,d+\frac13,2d+\frac16\right) &= \frac{ 2^{18d-\frac23} \Gamma\left(\frac13\right)^2 \Gamma\left(d+\frac56\right) \Gamma(d+1) \Gamma\left(2d+\frac12\right)^4 \Gamma\left(3d+\frac12\right) }{ 3^{6d}\pi^2 \Gamma\left(\frac56\right) \Gamma\left(2d+\frac13\right)^2 \Gamma\left(3d+\frac56\right) \Gamma\left(6d+\frac23\right) } \\ W\left(2d+\frac16;\frac23-d,\frac12,\frac12,d+\frac13,2d+\frac13\right) &= \frac{ 2^{8d+\frac13} \Gamma\left(\frac13\right)^2 \Gamma\left(2d+\frac16\right) \Gamma\left(2d+\frac12\right)^4 \Gamma\left(2d+\frac76\right) }{ 3^{6d}\sqrt{\pi} \Gamma\left(\frac56\right) \Gamma(2d) \Gamma\left(d+\frac16\right) \Gamma\left(d+\frac12\right) \Gamma\left(2d+\frac13\right) \Gamma\left(3d+\frac56\right)^2 } \\ W\left(3d-\frac16;\frac13,\frac12,d+\frac16,d+\frac13,3d\right) &= \frac{ 2^{16d+\frac23} \Gamma\left(\frac13\right)^2 \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac16\right) \Gamma\left(2d+\frac12\right)^3 \Gamma\left(3d+\frac12\right) }{ 3^{9d}\sqrt{\pi} \Gamma\left(\frac56\right) \Gamma\left(d+\frac16\right)^2 \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac13\right) \Gamma\left(6d+\frac23\right) } \\ W\left(3d;\frac23-d,d+\frac13,d+\frac13,2d+\frac16,2d+\frac13\right) &= \frac{ 2^{10d+\frac23} \Gamma\left(\frac13\right)^2 \Gamma\left(d+\frac56\right) \Gamma\left(2d+\frac23\right) \Gamma\left(3d+\frac12\right)^2 }{ 3^{9d}\sqrt{\pi} \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac13\right)^2 \Gamma(3d+1) } \\ W\left(5d-\frac12;d+\frac16,2d,2d+\frac16,3d,3d\right) &= \frac{ 2^{12d} \Gamma\left(\frac13\right)^2 \Gamma\left(\frac23-d\right) \Gamma\left(2d+\frac12\right)^2 \Gamma\left(3d+\frac13\right) \Gamma\left(3d+\frac12\right)^3 }{ 3^{9d}\pi \Gamma\left(d+\frac12\right)^2 \Gamma\left(2d+\frac13\right)^3 \Gamma\left(2d+\frac23\right) \Gamma\left(5d+\frac12\right) } \end{align}
$d\to\frac 16$における特殊値
多くの異なる和公式が得られたが, $d\to\frac 16$とすると得られる値は以下の6つである.
\begin{align} W\left(\frac23;\frac12,\frac12,\frac12,\frac23,\frac23\right) &= \frac{ 9\Gamma\left(\frac13\right)^{18} }{ 2^{12}\pi^{10} } \\ W\left(\frac12;\frac13,\frac12,\frac12,\frac12,\frac12\right) &= \frac{ 3\cdot 2^{\frac23}\Gamma\left(\frac13\right)^{12} }{ 64\pi^8 } \\ W\left(\frac56;\frac23,\frac23,\frac23,\frac23,\frac23\right) &= \frac{ 3\Gamma\left(\frac13\right)^{24} }{ 5\cdot 2^{13}\pi^{12} } \\ W\left(\frac13;\frac13,\frac13,\frac12,\frac12,\frac12\right) &= \frac{2\sqrt3}{\pi} \\ W\left(\frac12;\frac12,\frac12,\frac12,\frac12,\frac23\right) &= \frac{ \sqrt3\Gamma\left(\frac13\right)^6 }{ 2^{\frac23}\pi^5 } \\ W\left(\frac16;\frac13,\frac13,\frac13,\frac13,\frac13\right) &= \frac{ 64\sqrt3\pi^3 }{ 9\Gamma\left(\frac13\right)^6 } \end{align}
