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2
前の記事(
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
)で
三次変換公式から得られる積分
\begin{align}
&\int_0^1(x(1-x))^{2d-\frac 23}\F21{3d,d+\frac13}{2d+\frac 23}x\F21{3d,d+\frac 13}{2d+\frac 13}{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(\frac 23\right)\Gamma\left(d+\frac 12\right)^3\Gamma\left(2d+\frac 13\right)^2}
\end{align}
今回は, 別の
三次変換公式から得られる積分
\begin{align}
&\int_0^1(x(1-x^3))^{\frac{3c-1}2}\F21{c,c+\frac 13}{\frac{3c+5}6}{x^3}^2\,dx\\
&=\frac{\Gamma\left(\frac 13\right)^2\Gamma\left(\frac{3c+1}2\right)^2\Gamma\left(\frac{1-c}2\right)\Gamma\left(\frac{3c+1}6\right)}{3^{3c+1}\Gamma\left(\frac 23\right)\Gamma\left(\frac{c+1}2\right)^2\Gamma\left(c+\frac 13\right)^2}
\end{align}
から得られる${}_7F_6$和公式を求めたいと思う.
導出
以下の導出は,
\begin{align}
\F76{\frac 13,\frac 76,\frac 13,\frac 13,\frac 13,\frac 14,\frac 34}{\frac 16,1,1,1,\frac{13}{12},\frac 7{12}}1=\frac{9}{64}\frac{\Gamma\left(\frac 13\right)^9}{\pi^6}
\end{align}
を示した
Nader氏の記事
の方法を参考にしたものである.
まず, $t=x^3$として
\begin{align}
&\int_0^1(x(1-x^3))^{\frac{3c-1}2}\F21{c,c+\frac 13}{\frac{3c+5}6}{x^3}^2\,dx\\
&=\frac 13\int_0^1t^{\frac{3c-5}6}(1-t)^{\frac{3c-1}2}\F21{c,c+\frac 13}{\frac{3c+5}6}{t}^2\,dt
\end{align}
となる. Pfaff変換から
\begin{align}
\F21{c,c+\frac 13}{\frac{3c+5}6}{t}&=(1-t)^{-c}\F21{c,\frac{1-c}2}{\frac{3c+5}6}{-\frac{t}{1-t}}\,dt
\end{align}
であるから, $u=\frac{t}{1-t}$として
\begin{align}
&\int_0^1t^{\frac{3c-5}6}(1-t)^{\frac{3c-1}2}\F21{c,c+\frac 13}{\frac{3c+5}6}{t}^2\,dt\\
&=\int_0^1t^{\frac{3c-5}6}(1-t)^{-\frac{c+1}2}\F21{c,\frac{1-c}2}{\frac{3c+5}6}{-\frac{t}{1-t}}^2\,dt\\
&=\int_0^{\infty}u^{\frac{3c-5}6}(1+u)^{-\frac 23}\F21{c,\frac{1-c}2}{\frac{3c+5}6}{-u}^2\,du
\end{align}
となる. よって,
\begin{align}
f(u):=u^{\frac{3c-5}{12}}(1+u)^{-\frac 13}\F21{c,\frac{1-c}2}{\frac{3c+5}6}{-u}
\end{align}
とすると,
\begin{align}
&\int_0^1(x(1-x^3))^{\frac{3c-1}2}\F21{c,c+\frac 13}{\frac{3c+5}6}{x^3}^2\,dx\\
&=\frac 13\int_0^{\infty}f(u)^2\,du
\end{align}
と書き換えられることが分かった. ここで, $f$のMellin変換は$u=\frac{t}{1-t}$と戻して
\begin{align}
g(s)&:=\int_0^{\infty}u^{s-1}f(u)\,du\\
&=\int_0^{\infty}u^{s+\frac{3c-17}{12}}(1+u)^{-\frac 13}\F21{c,\frac{1-c}2}{\frac{3c+5}6}{-u}\,du\\
&=\int_0^1t^{s+\frac{3c-17}{12}}(1-t)^{-\frac{c+1}{4}-s}\F21{c,\frac{1-c}2}{\frac{3c+5}6}{-\frac{t}{1-t}}\,dt\\
&=\int_0^1t^{s+\frac{3c-17}{12}}(1-t)^{\frac{3c-1}4-s}\F21{c,c+\frac 13}{\frac{3c+5}6}{t}\,dt
\end{align}
となる. 項別積分により
\begin{align}
g(s)&=\sum_{0\leq n}\frac{\left(c,c+\frac 13\right)_n}{n!\left(\frac{3c+5}6\right)_n}\int_0^1t^{n+s+\frac{3c-17}{12}}(1-t)^{\frac{3c-1}4-s}\,dt\\
&=\sum_{0\leq n}\frac{\left(c,c+\frac 13\right)_n}{n!\left(\frac{3c+5}6\right)_n}\frac{\Gamma\left(\frac{3c-5}{12}+s+n\right)\Gamma\left(\frac{3c+3}4-s\right)}{\Gamma\left(c+\frac 13+n\right)}\\
&=\frac{\Gamma\left(\frac{3c-5}{12}+s\right)\Gamma\left(\frac{3c+3}4-s\right)}{\Gamma\left(c+\frac 13\right)}\F21{c,\frac{3c-5}{12}+s}{\frac{3c+5}6}1\\
&=\frac{\Gamma\left(\frac{3c-5}{12}+s\right)\Gamma\left(\frac{3c+3}4-s\right)}{\Gamma\left(c+\frac 13\right)}\frac{\Gamma\left(\frac{3c+5}6\right)\Gamma\left(\frac{5-3c}4-s\right)}{\Gamma\left(\frac{5-3c}6\right)\Gamma\left(\frac{c+5}4-s\right)}\\
&=\frac{\Gamma\left(\frac{3c+5}6\right)}{\Gamma\left(c+\frac 13\right)\Gamma\left(\frac{5-3c}6\right)}\frac{\Gamma\left(\frac{3c-5}{12}+s\right)\Gamma\left(\frac{3c+3}{4}-s\right)\Gamma\left(\frac{5-3c}4-s\right)}{\Gamma\left(\frac{c+5}4-s\right)}
\end{align}
が得られる. よって, 前の記事(
Bessel関数の4乗の積分
)の補題2から
\begin{align}
\int_0^{\infty}f(t)^2\,dt&=\frac 1{2\pi i}\int_{-i\infty}^{i\infty}g(s)g(1-s)\,ds\\
&=\frac 1{2\pi}\int_{-\infty}^{\infty}g\left(\frac 12+it\right)g\left(\frac 12-it\right)\,dt\\
&=\left(\frac{\Gamma\left(\frac{3c+5}6\right)}{\Gamma\left(c+\frac 13\right)\Gamma\left(\frac{5-3c}6\right)}\right)^2\frac 1{\pi}\int_0^{\infty}\left|\frac{\Gamma\left(\frac{3c+1}{12}+it\right)\Gamma\left(\frac{3c+1}4+it\right)\Gamma\left(\frac{3-3c}4+it\right)}{\Gamma\left(\frac{c+3}4+it\right)}\right|^2\,dt
\end{align}
を得る. ここで, Nassrallah-Rahman積分(
前の記事
の定理3)において$d=\frac 12,t=0$とすると
\begin{align}
\frac{\Gamma(it)\Gamma(-it)\Gamma\left(\frac 12+it\right)\Gamma\left(\frac 12-it\right)}{\Gamma(2it)\Gamma(-2it)}=4\pi
\end{align}
であるから,
\begin{align}
&\int_0^{\infty}\left|\frac{\Gamma(a+it)\Gamma(b+it)\Gamma(c+it)}{\Gamma(A+it)}\right|^2\,dt\\
&=\frac{\Gamma(a+b)\Gamma(a+c)\Gamma\left(a+\frac 12\right)\Gamma(b+c)\Gamma\left(b+\frac 12\right)\Gamma\left(c+\frac 12\right)}{2\Gamma\left(a+b+c+\frac 12\right)}\\
&\qquad\cdot\frac{\Gamma\left(A+a+c+\frac 12\right)\Gamma(a)\Gamma(c)\Gamma\left(\frac 12\right)}{\Gamma\left(a+c+\frac 12\right)\Gamma(A+a)\Gamma(A+c)\Gamma\left(A+\frac 12\right)}\\
&\qquad\cdot W\left(A+a+c-\frac 12;A-b,A,a+c,a+\frac 12,c+\frac 12\right)\\
&=\frac{2^{2-2a-2c}\pi^{\frac 32}\Gamma(a+b)\Gamma(a+c)\Gamma(b+c)\Gamma\left(b+\frac 12\right)\Gamma\left(A+a+c+\frac 12\right)\Gamma(2a)\Gamma(2c)}{2\Gamma\left(a+b+c+\frac 12\right)\Gamma\left(a+c+\frac 12\right)\Gamma(A+a)\Gamma(A+c)\Gamma\left(A+\frac 12\right)}\\
&\qquad\cdot W\left(A+a+c-\frac 12;A-b,A,a+c,a+\frac 12,c+\frac 12\right)\\
\end{align}
を得る. これを用いると,
\begin{align}
&\int_0^{\infty}\left|\frac{\Gamma\left(\frac{3c+1}{12}+it\right)\Gamma\left(\frac{3c+1}4+it\right)\Gamma\left(\frac{3-3c}4+it\right)}{\Gamma\left(\frac{c+3}4+it\right)}\right|^2\,dt\\
&=\frac{2^{c-\frac 23}\pi^{\frac 32}\Gamma\left(c+\frac 13\right)\Gamma\left(\frac{5-3c}6\right)\Gamma\left(\frac{3c+3}4\right)\Gamma\left(\frac{25-3c}{12}\right)\Gamma\left(\frac{3c+1}6\right)\Gamma\left(\frac{3-3c}2\right)}{\Gamma\left(\frac{3c+19}{12}\right)\Gamma\left(\frac{8-3c}6\right)\Gamma\left(\frac{3c+5}6\right)\Gamma\left(\frac{3-c}2\right)\Gamma\left(\frac{c+5}4\right)}\\
&\qquad\cdot W\left(\frac{13-3c}{12};\frac{1-c}2,\frac{c+3}4,\frac{5-3c}6,\frac{3c+7}{12},\frac{5-3c}4\right)
\end{align}
となる. ここで,
\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}
である. よって, 冒頭に述べた積分より
\begin{align}
&\frac{\Gamma\left(\frac 13\right)^2\Gamma\left(\frac{3c+1}2\right)^2\Gamma\left(\frac{1-c}2\right)\Gamma\left(\frac{3c+1}6\right)}{3^{3c+1}\Gamma\left(\frac 23\right)\Gamma\left(\frac{c+1}2\right)^2\Gamma\left(c+\frac 13\right)^2}\\
&=\frac 13\left(\frac{\Gamma\left(\frac{3c+5}6\right)}{\Gamma\left(c+\frac 13\right)\Gamma\left(\frac{5-3c}6\right)}\right)^2\frac 1{\pi}\\
&\qquad\cdot \frac{2^{c-\frac 23}\pi^{\frac 32}\Gamma\left(c+\frac 13\right)\Gamma\left(\frac{5-3c}6\right)\Gamma\left(\frac{3c+3}4\right)\Gamma\left(\frac{25-3c}{12}\right)\Gamma\left(\frac{3c+1}6\right)\Gamma\left(\frac{3-3c}2\right)}{\Gamma\left(\frac{3c+19}{12}\right)\Gamma\left(\frac{8-3c}6\right)\Gamma\left(\frac{3c+5}6\right)\Gamma\left(\frac{3-c}2\right)\Gamma\left(\frac{c+5}4\right)}\\
&\qquad\cdot W\left(\frac{13-3c}{12};\frac{1-c}2,\frac{c+3}4,\frac{5-3c}6,\frac{3c+7}{12},\frac{5-3c}4\right)
\end{align}
となるから,
\begin{align}
&W\left(\frac{13-3c}{12};\frac{1-c}2,\frac{c+3}4,\frac{5-3c}6,\frac{3c+7}{12},\frac{5-3c}4\right)\\
&=\frac{\Gamma\left(\frac 13\right)^2\Gamma\left(\frac{3c+1}2\right)^2\Gamma\left(\frac{1-c}2\right)\Gamma\left(\frac{3c+1}6\right)}{3^{3c+1}\Gamma\left(\frac 23\right)\Gamma\left(\frac{c+1}2\right)^2\Gamma\left(c+\frac 13\right)^2}\cdot 3\pi\left(\frac{\Gamma\left(\frac{3c+5}6\right)}{\Gamma\left(c+\frac 13\right)\Gamma\left(\frac{5-3c}6\right)}\right)^{-2}\\
&\qquad\cdot \frac{\Gamma\left(\frac{3c+19}{12}\right)\Gamma\left(\frac{8-3c}6\right)\Gamma\left(\frac{3c+5}6\right)\Gamma\left(\frac{3-c}2\right)\Gamma\left(\frac{c+5}4\right)}{2^{c-\frac 23}\pi^{\frac 32}\Gamma\left(c+\frac 13\right)\Gamma\left(\frac{5-3c}6\right)\Gamma\left(\frac{3c+3}4\right)\Gamma\left(\frac{25-3c}{12}\right)\Gamma\left(\frac{3c+1}6\right)\Gamma\left(\frac{3-3c}2\right)}\\
\end{align}
を得る. 整理すると以下を得る.
\begin{align} &W\left(\frac{13-3c}{12};\frac{1-c}2,\frac{c+3}4,\frac{5-3c}6,\frac{3c+7}{12},\frac{5-3c}4\right)\\ &=\frac{3^{\frac{3c-3}4}(c+1)(3c+7)\Gamma\left(\frac 13\right)^3\Gamma\left(\frac{3c+1}6\right)\Gamma\left(\frac{3c+5}6\right)\Gamma\left(\frac{8-3c}6\right)\Gamma\left(\frac{3-c}2\right)}{24\cdot 2^{2c+\frac 23}\pi\Gamma\left(\frac{3c+4}6\right)\Gamma\left(\frac{7-3c}6\right)\Gamma\left(\frac{25-3c}{12}\right)\Gamma\left(\frac{3c+11}{12}\right)} \end{align}
別の表示
定理1に 二項変換公式 , 三項変換公式 を適用して得られる${}_7F_6$和公式を以下にまとめる. ${}_7F_6$の二項変換公式で移り合うものを1つの定理の中にまとめる.
\begin{align} &W\left(\frac{13-3c}{12};\frac{1-c}{2},\frac{c+3}{4},\frac{5-3c}{6},\frac{3c+7}{12},\frac{5-3c}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c-7}{4}}\,\left(3c+7\right)\,\left(c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{8-3c}{6}\right)\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{3c+5}{6}\right)}{2^{\frac{6c+11}{3}}\,\pi\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{25-3c}{12}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{2c+1}{2};\frac{1}{2},\frac{2}{3},\frac{3c+1}{4},\frac{3c+3}{4},c\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\sqrt{\pi}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{c+3}{4}\right)\,\Gamma\left(\frac{6c+5}{6}\right)\,\Gamma\left(c+2\right)}{2^{\frac{6c+8}{3}}\,\sin\left(\frac{\pi\left(3c+1\right)}{6}\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+7}{12}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)\,\Gamma\left(\frac{2c+3}{2}\right)}\\ &W\left(\frac{c+1}{2};\frac{1-c}{2},\frac{2}{3},\frac{c+1}{4},\frac{c+3}{4},c\right)=\frac{2^{\frac{2}{3}}\,\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{c+2}{2}\right)}{3\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{2-c}{2};\frac{3-3c}{4},\frac{5-3c}{4},\frac{1-c}{2},\frac{1}{2},\frac{2}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\left(c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{8-3c}{6}\right)\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{3c+1}{12}\right)\,\Gamma\left(\frac{c+3}{4}\right)\,\Gamma\left(\frac{3c+1}{6}\right)}{2^{\frac{3c+25}{6}}\,\pi^{\frac{3}{2}}\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{4-c}{2}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{c+3}{4};\frac{3-3c}{4},\frac{2}{3},1,\frac{c+1}{4},\frac{3c+1}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(c+2\right)}{2^{\frac{6c+2}{3}}\,\sqrt{\pi}\,\left(c+3\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+5}{4}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{6c+5}{6};\frac{5}{6},1,\frac{3c+1}{4},\frac{3c+3}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\left(3c+7\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{1-c}{2}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(c+2\right)}{2^{\frac{6c+11}{3}}\,\sqrt{\pi}\,\left(6c+5\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+5}{4}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{3c+5}{6};\frac{5-3c}{6},1,\frac{c+1}{4},\frac{c+3}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\left(3c+7\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(c+2\right)}{2^{\frac{6c+8}{3}}\,\sqrt{\pi}\,\left(3c+5\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+5}{4}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{c+5}{4};\frac{5-3c}{4},\frac{2}{3},1,\frac{c+3}{4},\frac{3c+3}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c-7}{4}}\,\left(3c+7\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{3c+1}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(c+2\right)}{2^{\frac{6c+14}{3}}\,\sqrt{\pi}\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+9}{4}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{3c+2}{6};\frac{1-c}{2},\frac{1}{2},\frac{3c+1}{12},\frac{3c+7}{12},c\right)\notag\\ &\quad=\frac{\left(c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{8-3c}{6}\right)\,\Gamma\left(\frac{6c+5}{6}\right)}{2^{\frac{3c+5}{3}}\,\sqrt{3}\,\pi\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+8}{6}\right)}\\ &W\left(\frac{8-3c}{6};\frac{3-3c}{4},\frac{5-3c}{4},\frac{5-3c}{6},\frac{5}{6},1\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\left(3c+7\right)\,\left(c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(c\right)}{2^{\frac{6c+11}{3}}\,\sqrt{\pi}\,\left(8-3c\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+5}{4}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{15c+1}{12};\frac{3c+1}{12},\frac{c+1}{4},\frac{3c+1}{4},c,\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-1}{2}}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{5-3c}{4}\right)\,\Gamma\left(\frac{c+3}{4}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{3c+5}{6}\right)\,\Gamma\left(\frac{6c+5}{6}\right)\,\Gamma\left(c+1\right)}{2^{\frac{3c+7}{6}}\,\pi^{\frac{5}{2}}\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{15c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{2}\right)}\\ &W\left(\frac{3c+7}{12};\frac{3-3c}{4},\frac{1}{2},\frac{5}{6},\frac{3c+1}{12},\frac{3c+1}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c-1}{2}}\,\left(3c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{8-3c}{6}\right)\,\Gamma\left(\frac{3c+1}{12}\right)^{2}\,\Gamma\left(\frac{c+3}{4}\right)^{2}\,\Gamma\left(\frac{6c+5}{6}\right)}{8\,\pi^{\frac{5}{2}}\,\left(3c+7\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{2}\right)}\\ &W\left(\frac{3c+4}{6};\frac{5-3c}{6},\frac{5}{6},\frac{3c+1}{12},\frac{3c+7}{12},\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-5}{4}}\,\left(3c+7\right)\,\Gamma\left(\frac{1}{3}\right)^{2}\,\Gamma\left(\frac{8-3c}{6}\right)\,\Gamma\left(\frac{3c+5}{12}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{6c+5}{6}\right)}{2^{\frac{9c-1}{6}}\,\sqrt{\pi}\,\left(3c+4\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{c+1}{4}\right)\,\Gamma\left(\frac{3c+4}{6}\right)^{2}\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{7-3c}{12};\frac{3-3c}{4},\frac{1-c}{2},\frac{5-3c}{6},\frac{3c+1}{12},\frac{c+1}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c-1}{2}}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{8-3c}{6}\right)\,\Gamma\left(\frac{3-c}{2}\right)\,\Gamma\left(\frac{c+3}{4}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{3c+5}{6}\right)}{2^{\frac{6c+2}{3}}\,\pi^{2}\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{19-3c}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)}\\ &W\left(\frac{15c+7}{12};\frac{3c+7}{12},\frac{c+3}{4},\frac{3c+3}{4},c,\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-7}{4}}\,\left(3c+7\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-3c}{4}\right)\,\Gamma\left(\frac{3c+5}{12}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{6c+5}{6}\right)\,\Gamma\left(c+2\right)}{2^{\frac{9c+23}{6}}\,\pi^{\frac{3}{2}}\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)\,\Gamma\left(\frac{15c+19}{12}\right)}\\ &W\left(\frac{3c+13}{12};\frac{5-3c}{4},\frac{1}{2},\frac{5}{6},\frac{3c+7}{12},\frac{3c+3}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c+5}{4}}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{8-3c}{6}\right)\,\Gamma\left(\frac{c+5}{4}\right)\,\Gamma\left(\frac{3c+19}{12}\right)\,\Gamma\left(\frac{6c+5}{6}\right)}{2^{\frac{3c-1}{2}}\,\pi\,\left(3c+13\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+5}{4}\right)}\end{align}
\begin{align} &W\left(\frac{3-c}{4};1-c,\frac{5-3c}{12},\frac{1}{2},1,\frac{c+1}{2}\right)\notag\\ &\quad=\frac{ 2^{\frac{4}{3}-c}3^{\frac{3c-3}{2}} \cos\left(\frac{\pi(3c+1)}{12}\right) \tan\left(\frac{\pi(c+1)}{4}\right) \Gamma\left(\frac13\right)^3 \Gamma\left(\frac{5-3c}{4}\right) \Gamma\left(\frac{3c+1}{12}\right) \Gamma\left(\frac{3c+1}{6}\right) }{ (c-3)\cos\left(\frac{\pi(3c+1)}{4}\right) \Gamma\left(\frac{1-3c}{6}\right) \Gamma\left(\frac{5-3c}{12}\right) \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c-3}{4}\right) }\\ &W\left(\frac{9c+1}{12};\frac{1}{3},\frac{1}{2},\frac{c+1}{2},\frac{3c-1}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{ 2^{\frac{3c-1}{6}}3^{\frac{3c-3}{2}}\sqrt{\pi} \Gamma\left(\frac13\right)^3 \Gamma\left(\frac{3-3c}{2}\right) \Gamma\left(\frac{c+3}{4}\right) \Gamma\left(\frac{3c+1}{6}\right) \Gamma\left(\frac{9c+7}{12}\right) \Gamma\left(\frac{3c+3}{4}\right) }{ \Gamma\left(\frac{5-3c}{12}\right)^2 \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{7-3c}{6}\right) \Gamma\left(\frac{c+1}{4}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c+1}{4}\right) \Gamma\left(\frac{9c+13}{12}\right) }\\ &W\left(\frac{7-3c}{12};1-c,\frac{1-c}{4},\frac{1}{3},\frac{5}{6},\frac{c+1}{2}\right)\notag\\ &\quad=-\frac{ 2^{\frac13-c}3^{\frac{3c-5}{2}} \cos\left(\frac{\pi(3c+1)}{12}\right) \Gamma\left(\frac13\right)^3 \Gamma\left(\frac{13-9c}{12}\right) \Gamma\left(\frac{5-3c}{4}\right) \Gamma\left(\frac{3c+1}{6}\right) \Gamma\left(\frac{9c+7}{12}\right) }{ \cos\left(\frac{\pi(c+1)}{4}\right) \Gamma\left(\frac{5-3c}{12}\right) \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{7-3c}{6}\right) \Gamma\left(\frac{19-3c}{12}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c-3}{4}\right) }\\ &W\left(\frac{9c+7}{12};\frac{5}{6},1,\frac{c+1}{2},\frac{3c+1}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=-\frac{ 2^{\frac73-c}3^{\frac{3c-3}{2}} \cos\left(\frac{\pi(3c+1)}{12}\right) \tan\left(\frac{\pi(c+1)}{4}\right) \Gamma\left(\frac13\right)^3 \Gamma\left(\frac{5-3c}{4}\right) \Gamma\left(\frac{3c+13}{12}\right) \Gamma\left(\frac{3c+1}{6}\right) }{ (9c+7)\cos\left(\frac{\pi(3c+1)}{4}\right) \Gamma\left(\frac{5-3c}{12}\right) \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{7-3c}{6}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c-3}{4}\right) }\\ &W\left(0;\frac{1-3c}{4},\frac{5-3c}{12},\frac{1}{2},\frac{c+1}{4},\frac{3c-1}{4}\right)\notag\\ &\quad=\frac{ 2^{\frac23-c}3^{\frac{3c}{2}}\pi^3 \Gamma\left(\frac{5-3c}{4}\right) \Gamma\left(\frac{c+3}{4}\right) \Gamma\left(\frac{3c+1}{6}\right) }{ \cos\left(\frac{\pi(3c+1)}{4}\right) \Gamma\left(\frac{1-3c}{6}\right) \Gamma\left(\frac{5-3c}{12}\right)^2 \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{c+1}{4}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c+1}{4}\right) }\\ &W\left(-\frac{1}{6};\frac{1-3c}{4},\frac{1-c}{4},\frac{1}{3},\frac{3c+1}{12},\frac{3c-1}{4}\right)\notag\\ &\quad=\frac{ 2^{\frac13-c}3^{\frac{3c-1}{2}}\sqrt{\pi} \Gamma\left(\frac13\right)^2 \Gamma\left(\frac{13-9c}{12}\right) \Gamma\left(\frac{5-3c}{4}\right) \Gamma\left(\frac{c+3}{4}\right) \Gamma\left(\frac{3c+1}{6}\right) \Gamma\left(\frac{9c+7}{12}\right) }{ \Gamma\left(\frac56\right) \Gamma\left(\frac{5-3c}{12}\right)^2 \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{7-3c}{6}\right) \Gamma\left(\frac{c+1}{4}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c+1}{4}\right) }\\ &W\left(\frac{1}{2};\frac{3-3c}{4},\frac{5-3c}{12},1,\frac{c+1}{4},\frac{3c+1}{4}\right)\notag\\ &\quad=-\frac{ 2^{\frac13-c}3^{\frac{3c-1}{2}} \cos\left(\frac{\pi(3c+1)}{12}\right) \tan\left(\frac{\pi(c+1)}{4}\right) \Gamma\left(\frac13\right)^3 \Gamma\left(\frac{5-3c}{4}\right) \Gamma\left(\frac{3c+13}{12}\right) \Gamma\left(\frac{3c+1}{6}\right) }{ \cos\left(\frac{\pi(3c+1)}{4}\right) \Gamma\left(\frac{1-3c}{6}\right) \Gamma\left(\frac{5-3c}{12}\right) \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c-3}{4}\right) }\\ &W\left(\frac{1}{3};\frac{3-3c}{4},\frac{1-c}{4},\frac{5}{6},\frac{3c+1}{12},\frac{3c+1}{4}\right)\notag\\ &\quad=-\frac{ 2^{\frac13-c}3^{\frac{3c-1}{2}} \cos\left(\frac{\pi(3c+1)}{12}\right) \tan\left(\frac{\pi(c+1)}{4}\right) \Gamma\left(\frac13\right) \Gamma\left(\frac{13-9c}{12}\right) \Gamma\left(\frac{5-3c}{4}\right) \Gamma\left(\frac{3c+13}{12}\right) \Gamma\left(\frac{3c+1}{6}\right) \Gamma\left(\frac{9c+7}{12}\right) }{ \Gamma\left(\frac{5-3c}{12}\right) \Gamma\left(\frac{2-c}{2}\right) \Gamma\left(\frac{7-3c}{6}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c-3}{4}\right) }\\ &W\left(\frac{c}{2};\frac{1-c}{4},\frac{5-3c}{12},\frac{c+1}{2},\frac{3c-1}{4},\frac{3c+1}{4}\right)\notag\\ &\quad=-\frac{ 2^{\frac13-c}3^{\frac{3c-1}{2}}\pi^{\frac32} \sin\left(\frac{\pi c}{2}\right) \cos\left(\frac{\pi(3c+1)}{12}\right) \Gamma\left(\frac13\right)^2 \Gamma\left(\frac{9c+7}{12}\right) }{ c\sin\left(\frac{\pi(3c+1)}{6}\right) \cos\left(\frac{\pi(c+1)}{4}\right) \cos\left(\frac{\pi(3c+1)}{4}\right) \Gamma\left(\frac{1-3c}{6}\right) \Gamma\left(\frac{5-3c}{12}\right) \Gamma\left(\frac{c+1}{4}\right) \Gamma\left(\frac{3c+4}{6}\right) \Gamma\left(\frac{3c-3}{4}\right) }\\ &W\left(\frac{1-2c}{2};1-c,\frac{1-3c}{4},\frac{3-3c}{4},\frac{1-c}{4},\frac{5-3c}{12}\right)\notag\\ &\quad=\frac{ 2^{-3c-\frac13}3^{\frac{3c-1}{2}} \cos\left(\frac{\pi(3c+1)}{12}\right) \Gamma\left(\frac13\right)^2 \Gamma\left(\frac{13-9c}{12}\right) \Gamma\left(\frac{5-3c}{4}\right)^2 \Gamma\left(\frac{3-c}{2}\right) \Gamma\left(\frac{3c+1}{6}\right)^2 }{ \pi\cos\left(\frac{\pi(c+1)}{4}\right) \Gamma\left(\frac{5-3c}{12}\right) \Gamma\left(\frac{7-3c}{6}\right) \Gamma\left(2-2c\right) \Gamma\left(\frac{c+1}{4}\right) \Gamma\left(\frac{3c+1}{4}\right) }\end{align}
\begin{align} &W\left(\frac{3c+13}{12};\frac{5-3c}{6},\frac{1}{2},1,\frac{c+3}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\left(3c+7\right)\,\left(1-c\right)\,\sqrt{\pi}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-3c}{4}\right)^{2}\,\Gamma\left(\frac{3-c}{4}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{3c+5}{4}\right)}{2^{\frac{6c+5}{3}}\,\left(3c+1\right)\,\cos\left(\frac{\pi\left(c-1\right)}{4}\right)\,\Gamma\left(1-c\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{-3c-1}{12}\right)^{2}\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+25}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{9c+7}{12};\frac{1}{3},\frac{1}{2},\frac{c+1}{2},\frac{3c+1}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{2^{c-3}\,3^{\frac{3c-11}{4}}\,\pi\,\left(3c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{5-3c}{2}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{3c+1}{4}\right)\,\Gamma\left(\frac{9c+13}{12}\right)}{\Gamma\left(\frac{2-c}{2}\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{11-3c}{12}\right)^{2}\,\Gamma\left(\frac{c+3}{4}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{9c+19}{12}\right)}\\ &W\left(\frac{13-3c}{12};1-c,\frac{3-c}{4},\frac{1}{3},\frac{5}{6},\frac{c+1}{2}\right)\notag\\ &\quad=\frac{3^{\frac{3c+1}{4}}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-3c}{4}\right)\,\Gamma\left(\frac{19-9c}{12}\right)\,\Gamma\left(\frac{5-c}{2}\right)\,\Gamma\left(\frac{3-c}{4}\right)\,\Gamma\left(\frac{9c+13}{12}\right)}{2^{\frac{3c-5}{2}}\,\left(3c+1\right)^{2}\,\Gamma\left(1-c\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{-3c-1}{12}\right)^{2}\,\Gamma\left(\frac{25-3c}{12}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{9c+13}{12};\frac{5}{6},1,\frac{c+1}{2},\frac{3c+3}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=-\frac{3^{\frac{3c-3}{4}}\,\left(3c+7\right)\,\left(3-c\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-3c}{4}\right)^{2}\,\Gamma\left(\frac{3-c}{4}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{3c+5}{4}\right)\,\sin\left(\frac{\pi\left(3c+1\right)}{12}\right)}{2^{\frac{6c-1}{3}}\,\sqrt{\pi}\,\left(9c+13\right)\,\left(3c+1\right)\,\cos\left(\frac{\pi\left(c-1\right)}{4}\right)\,\Gamma\left(1-c\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{-3c-1}{12}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(1;\frac{3-3c}{4},\frac{11-3c}{12},\frac{1}{2},\frac{c+3}{4},\frac{3c+1}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c-5}{4}}\,\pi^{\frac{7}{2}}\,\left(1-c\right)\,\left(3c+1\right)\,\Gamma\left(\frac{7-3c}{4}\right)\,\Gamma\left(\frac{3c+13}{12}\right)}{2^{\frac{3c+19}{6}}\,\sin\left(\frac{\pi\left(3c+1\right)}{4}\right)\,\Gamma\left(\frac{2-c}{2}\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{11-3c}{12}\right)^{2}\,\Gamma\left(\frac{c+3}{4}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{5}{6};\frac{3-3c}{4},\frac{3-c}{4},\frac{1}{3},\frac{3c+7}{12},\frac{3c+1}{4}\right)\notag\\ &\quad=\frac{3^{\frac{3c-7}{4}}\,\pi\,\left(3c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{2}\,\Gamma\left(\frac{19-9c}{12}\right)\,\Gamma\left(\frac{7-3c}{4}\right)\,\Gamma\left(\frac{3c+1}{12}\right)\,\Gamma\left(\frac{9c+13}{12}\right)}{5\,2^{\frac{c+3}{2}}\,\Gamma\left(\frac{5}{6}\right)\,\Gamma\left(\frac{2-c}{2}\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{11-3c}{12}\right)^{2}\,\Gamma\left(\frac{c+3}{4}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{3}{2};\frac{5-3c}{4},\frac{11-3c}{12},1,\frac{c+3}{4},\frac{3c+3}{4}\right)\notag\\ &\quad=-\frac{3^{\frac{3c-3}{4}}\,\left(3c+7\right)\,\left(1-c\right)\,\left(3-c\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3-3c}{4}\right)^{2}\,\Gamma\left(\frac{3-c}{4}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{3c+5}{4}\right)\,\sin\left(\frac{\pi\left(3c+1\right)}{12}\right)}{2^{\frac{6c+8}{3}}\,\sqrt{\pi}\,\left(3c+1\right)\,\cos\left(\frac{\pi\left(c-1\right)}{4}\right)\,\Gamma\left(1-c\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{-3c-1}{12}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{c+2}{2};\frac{3-c}{4},\frac{11-3c}{12},\frac{c+1}{2},\frac{3c+1}{4},\frac{3c+3}{4}\right)\notag\\ &\quad=\frac{2^{\frac{3c-37}{6}}\,3^{\frac{3c-3}{4}}\,\pi^{\frac{3}{2}}\,\left(1-c\right)\,\left(3-c\right)\,\left(3c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{2}\,\Gamma\left(\frac{5-3c}{12}\right)\,\Gamma\left(\frac{9c+13}{12}\right)}{\left(c+2\right)\,\sin\left(\frac{\pi\left(3c+1\right)}{4}\right)\,\sin\left(\frac{\pi\left(c+1\right)}{4}\right)\,\Gamma\left(\frac{2-c}{2}\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{11-3c}{12}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(c+1\right)}\\ &W\left(\frac{4}{3};\frac{5-3c}{4},\frac{3-c}{4},\frac{5}{6},\frac{3c+7}{12},\frac{3c+3}{4}\right)\notag\\ &\quad=-\frac{3^{\frac{3c+5}{4}}\,\sqrt{\pi}\,\left(3c+7\right)\,\left(3-c\right)\,\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{3-3c}{4}\right)\,\Gamma\left(\frac{19-9c}{12}\right)\,\Gamma\left(\frac{3-c}{4}\right)\,\Gamma\left(\frac{3c+1}{6}\right)\,\Gamma\left(\frac{9c+13}{12}\right)\,\sin\left(\frac{\pi\left(3c+1\right)}{12}\right)}{2^{\frac{6c+14}{3}}\,\left(3c+1\right)\,\cos\left(\frac{\pi\left(c-1\right)}{4}\right)\,\Gamma\left(1-c\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{-3c-1}{12}\right)\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{6c+5}{6};\frac{3c+7}{12},\frac{c+3}{4},\frac{3c+1}{4},\frac{3c+3}{4},\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-3}{4}}\,\pi^{2}\,\left(3c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{2}\,\Gamma\left(\frac{3c+13}{6}\right)\,\Gamma\left(\frac{9c+13}{12}\right)}{2^{\frac{6c+14}{3}}\,\sin\left(\frac{\pi\left(3c+1\right)}{4}\right)\,\sin\left(\frac{\pi\left(c+1\right)}{4}\right)\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{11-3c}{12}\right)^{2}\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{6c+11}{6}\right)}\end{align}
\begin{align} &W\left(\frac{7c+1}{4};\frac{c+3}{4},\frac{3c+1}{4},\frac{3c+3}{4},c,\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c}{2}}\,\pi\,\cos\left(\frac{\pi c}{2}\right)\,\left(3c+1\right)\,\left(3c+7\right)\,\left(c+1\right)\,\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{3-3c}{2}\right)\,\Gamma\left(\frac{9c+11}{12}\right)\,\Gamma\left(\frac{3c+1}{2}\right)\,\Gamma\left(2c+1\right)}{2^{\frac{15c+29}{6}}\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{11-3c}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+11}{4}\right)\,\Gamma\left(\frac{7c+5}{4}\right)}\\ &W\left(\frac{2c+1}{2};\frac{1}{2},1,\frac{c+1}{4},\frac{c+3}{4},\frac{3c+1}{3}\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{c+2}{2}\right)\,\Gamma\left(\frac{3c+7}{6}\right)}{2^{\frac{4}{3}}\,\pi\,\left(2c+1\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{6c+1}{6};\frac{1}{6},\frac{2}{3},\frac{c+1}{4},\frac{c+3}{4},c\right)=\frac{\sqrt{6}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{9c+5}{6}\right)\,\Gamma\left(2c+1\right)}{2^{\frac{18c+13}{6}}\,3\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{6c+7}{6}\right)\,\Gamma\left(\frac{3c+1}{2}\right)}\\ &W\left(\frac{3c}{2};\frac{1}{2},\frac{c+1}{2},\frac{3c-1}{4},\frac{3c+1}{4},\frac{3c+1}{3}\right)=\frac{\pi\,\Gamma\left(\frac{2c+1}{2}\right)\,\Gamma\left(\frac{3c+3}{2}\right)}{2^{c}\,\sqrt{3}\,\Gamma\left(\frac{c+1}{2}\right)^{2}\,\Gamma\left(\frac{3c+2}{2}\right)}\\ &W\left(\frac{3c+1}{2};1,\frac{c+1}{2},\frac{3c+1}{4},\frac{3c+3}{4},\frac{3c+1}{3}\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{c+2}{2}\right)\,\Gamma\left(\frac{3c+7}{6}\right)}{2^{\frac{4}{3}}\,\pi\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{9c-2}{6};\frac{1}{6},\frac{3c+1}{6},\frac{3c-1}{4},\frac{3c+1}{4},c\right)=\frac{\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{2c+1}{2}\right)\,\Gamma\left(\frac{9c+5}{6}\right)}{2^{\frac{3c+1}{3}}\,\Gamma\left(\frac{c+1}{2}\right)^{2}\,\Gamma\left(\frac{9c+4}{6}\right)}\\ &W\left(\frac{9c+1}{6};\frac{2}{3},\frac{3c+1}{6},\frac{3c+1}{4},\frac{3c+3}{4},c\right)=\frac{\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{c+2}{2}\right)\,\Gamma\left(\frac{3c+7}{6}\right)\,\Gamma\left(\frac{9c+5}{6}\right)}{2^{\frac{1}{3}}\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{9c+7}{6}\right)}\\ &W\left(\frac{7c-1}{4};\frac{c+1}{4},\frac{3c-1}{4},\frac{3c+1}{4},c,\frac{3c+1}{3}\right)\notag\\ &\quad=\frac{3^{\frac{3c-2}{2}}\,\pi\,\cos\left(\frac{\pi c}{2}\right)\,\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{3-3c}{2}\right)\,\Gamma\left(\frac{9c+5}{12}\right)\,\Gamma\left(\frac{3c+1}{2}\right)\,\Gamma\left(2c+1\right)}{2^{\frac{15c-7}{6}}\,\Gamma\left(\frac{7-3c}{6}\right)\,\Gamma\left(\frac{5-3c}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)\,\Gamma\left(\frac{3c+1}{4}\right)\,\Gamma\left(\frac{7c+3}{4}\right)}\\ &W\left(\frac{3c+4}{6};\frac{5-3c}{12},\frac{11-3c}{12},\frac{1}{2},1,\frac{c+1}{2}\right)=\frac{\left(3c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{c}{2}\right)\,\Gamma\left(\frac{3c+7}{6}\right)}{2^{\frac{13}{3}}\,\sqrt{3}\,\pi\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+10}{6}\right)}\\ &W\left(\frac{9c-1}{12};\frac{5-3c}{12},\frac{1}{6},\frac{1}{2},\frac{c+1}{4},\frac{3c-1}{4}\right)=\frac{\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{9c+5}{12}\right)\,\Gamma\left(\frac{3c+3}{4}\right)\,\Gamma\left(\frac{2c+1}{2}\right)}{2^{\frac{3c+4}{3}}\,\sqrt{3}\,\sqrt{\pi}\,\Gamma\left(\frac{c+1}{2}\right)^{2}\,\Gamma\left(\frac{3c+1}{4}\right)\,\Gamma\left(\frac{9c+11}{12}\right)}\\ &W\left(\frac{9c+5}{12};\frac{5-3c}{12},\frac{2}{3},1,\frac{c+1}{4},\frac{3c+1}{4}\right)=\frac{2^{\frac{2}{3}}\,\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{c+2}{2}\right)\,\Gamma\left(\frac{3c+7}{6}\right)}{\pi\,\left(9c+5\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{3c+2}{6};\frac{5-3c}{12},\frac{11-3c}{12},\frac{1}{6},\frac{2}{3},\frac{3c+1}{6}\right)=\frac{\sqrt{6}\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{3c+7}{6}\right)\,\Gamma\left(\frac{3c+1}{3}\right)\,\Gamma\left(\frac{9c+5}{6}\right)}{2^{\frac{6c+13}{6}}\,3\,\sqrt{\pi}\,\Gamma\left(\frac{c+1}{2}\right)^{2}\,\Gamma\left(\frac{3c+8}{6}\right)\,\Gamma\left(\frac{3c+1}{2}\right)}\\ &W\left(\frac{9c+5}{12};\frac{11-3c}{12},\frac{1}{6},\frac{1}{2},\frac{c+3}{4},\frac{3c+1}{4}\right)=\frac{\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{9c+11}{12}\right)\,\Gamma\left(\frac{3c+5}{4}\right)\,\Gamma\left(\frac{2c+1}{2}\right)}{2^{\frac{3c+4}{3}}\,\sqrt{3}\,\sqrt{\pi}\,\Gamma\left(\frac{c+1}{2}\right)^{2}\,\Gamma\left(\frac{3c+3}{4}\right)\,\Gamma\left(\frac{9c+17}{12}\right)}\\ &W\left(\frac{9c+11}{12};\frac{11-3c}{12},\frac{2}{3},1,\frac{c+3}{4},\frac{3c+3}{4}\right)\notag\\ &\quad=-\frac{2^{\frac{2}{3}}\,\sqrt{3}\,\left(3c+1\right)\,\Gamma\left(\frac{1}{3}\right)^{3}\,\Gamma\left(\frac{c+2}{2}\right)\,\Gamma\left(\frac{3c+7}{6}\right)}{\pi\,\left(1-3c\right)\,\left(9c+11\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{3c+4}{6}\right)}\\ &W\left(\frac{15c-1}{12};\frac{5-3c}{12},\frac{3c+1}{6},\frac{c+1}{2},\frac{3c-1}{4},\frac{3c+1}{4}\right)\notag\\ &\quad=\frac{\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{3c+7}{6}\right)\,\Gamma\left(\frac{9c+5}{12}\right)\,\Gamma\left(\frac{3c+7}{4}\right)}{3\,\Gamma\left(\frac{c+5}{4}\right)\,\Gamma\left(\frac{c+1}{2}\right)\,\Gamma\left(\frac{15c+11}{12}\right)}\\ &W\left(\frac{15c+5}{12};\frac{11-3c}{12},\frac{3c+1}{6},\frac{c+1}{2},\frac{3c+1}{4},\frac{3c+3}{4}\right)\notag\\ &\quad=\frac{\sqrt{\pi}\,\Gamma\left(\frac{1}{3}\right)\,\Gamma\left(\frac{3c+13}{12}\right)\,\Gamma\left(\frac{9c+11}{12}\right)\,\Gamma\left(\frac{3c+3}{2}\right)}{2^{\frac{3c-1}{6}}\,3^{\frac{3c+1}{4}}\,\Gamma\left(\frac{3c+11}{12}\right)\,\Gamma\left(\frac{c+1}{2}\right)^{2}\,\Gamma\left(\frac{15c+17}{12}\right)}\end{align}
$c=\frac 13$における特殊値
上の和公式から得られる$c=\frac 13$における特殊値も以下にまとめておく.
\begin{align}
&W\left(1;\frac{1}{3},\frac{2}{3},\frac{2}{3},\frac{5}{6},1\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{9}}{324 \pi^{3}}\\
&W\left(\frac{5}{6};\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{2}{3},1\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{9}}{360 \pi^{3}}\\
&W\left(\frac{7}{6};\frac{1}{2},\frac{2}{3},\frac{5}{6},1,1\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{9}}{252 \pi^{3}}\\
&W\left(\frac{1}{2};\frac{1}{6},\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{2}{3}\right)=\frac{\Gamma\left(\frac{1}{3}\right)^{9}}{72 \pi^{4}}\\
&W\left(\frac{2}{3};\frac{1}{6},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{5}{6}\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{9}}{384 \pi^{3}}\\
&W\left(\frac{5}{6};\frac{1}{6},\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{5}{6}\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{12}}{720 \pi^{5}}
\\
&W\left(\frac{7}{6};\frac{1}{2},\frac{2}{3},\frac{2}{3},\frac{5}{6},1\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{9}}{336 \pi^{3}}\\
&W\left(\frac{5}{6};\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{2}{3},\frac{2}{3}\right)=\frac{\Gamma\left(\frac{1}{3}\right)^{18}}{5120 \pi^{8}}\\
&W\left(1;\frac{1}{3},\frac{2}{3},\frac{2}{3},\frac{2}{3},\frac{5}{6}\right)=\frac{\Gamma\left(\frac{1}{3}\right)^{12}}{1296 \pi^{4}}\\
&W\left(\frac{3}{2};\frac{5}{6},\frac{5}{6},1,1,1\right)=\frac{\sqrt{3}\,\Gamma\left(\frac{1}{3}\right)^{9}}{216 \pi^{3}}
\end{align}
これらにはNader氏によって示された
\begin{align}
\F76{\frac 13,\frac 76,\frac 13,\frac 13,\frac 13,\frac 14,\frac 34}{\frac 16,1,1,1,\frac{13}{12},\frac 7{12}}1=\frac{9}{64}\frac{\Gamma\left(\frac 13\right)^9}{\pi^6}
\end{align}
が含まれていないが, おそらくNassrallah-Rahman型積分の展開の仕方が本質的に異なることが理由だと思われる.
