一个超几何函数$_3F_2$的积分

\[\Large\displaystyle \int_0^\infty{_3F_2}\left(\begin{array}c\dfrac58,\dfrac58,\dfrac98\\\dfrac12,\dfrac{{13}}8\end{array}\middle|\ {-x}\right)^2\frac{\mathrm{d}x}{\sqrt x} \]


\(\Large\mathbf{Solution:}\)
Looks like we have a more general result:

\[\begin{align*} I(a)&=\int_0^\infty{_3F_2}\left(\begin{array}ca,a,a+\dfrac12\\\dfrac12,a+1\end{array}\middle|\ {-x}\right)^2\frac{\mathrm{d}x}{\sqrt x}\\ &=\frac{2^{4a-2}a^2}{\Gamma(2a)^2}G^{3~3}_{4~4}\left(1\middle|\begin{array}c\dfrac12,1,1;2a+\dfrac12\\2a-\dfrac12,2a-\dfrac12,2a;0\end{array}\right)\\ &=\frac{2^{4a-2}a^2}{\Gamma(2a)^2}G^{4~4}_{6~6}\left(1\middle|\begin{array}c\dfrac12,\dfrac12,1,1;2a,2a+\dfrac12\\2a-\dfrac12,2a-\dfrac12,2a,2a;0,\dfrac12\end{array}\right)\\ &=\frac{4\pi a^2}{\Gamma(2a)^2}G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;4a\\4a-1,4a-1;0\end{array}\right) \end{align*}\]

Here Mathematica gives when \(a=\dfrac58\),

\[\begin{align*} &G^{2~2}_{3~3}\left(z\middle|\begin{array}c1,1;\dfrac52\\\dfrac32,\dfrac32;0\end{array}\right)\\ &=\frac{2\sqrt\pi}{3}\left(\ln(\sqrt z+\sqrt{z+1})+z^{3/2}\ln(1+\sqrt{z+1})-z^{3/2}\ln(\sqrt z)-\sqrt{z}\sqrt{z+1})\right) \end{align*}\]

So

\[G^{2~2}_{3~3}\left(1\middle|\begin{array}c1,1;\dfrac52\\\dfrac32,\dfrac32;0\end{array}\right)=\frac{2\sqrt\pi}{3}(2\ln(1+\sqrt2)-\sqrt2) \]

Now we have:

\[\large\boxed{\displaystyle \int_0^\infty{_3F_2}\left(\begin{array}c\dfrac58,\dfrac58,\dfrac98\\\dfrac12,\dfrac{{13}}8\end{array}\middle|\ {-x}\right)^2\frac{\mathrm{d}x}{\sqrt x}=\color{Blue} {\frac{50\,\pi^{3/2}}{3\,\Gamma^2\left(\dfrac14\right)}\Big(\ln\left(3+\sqrt8\right)-\sqrt2\Big)}} \]

posted @ 2016-05-15 20:51  Renascence_5  阅读(585)  评论(0编辑  收藏