<s>[INST] Integral of powers of Bessel function from 0 to infinity Let $J_m(x)$ be the Bessel function of the first kind with order $m$. I was experimenting with the following integral on Wolfram Alpha
$$ \int_0^\infty J_m(x)^4\, dx, $$
and it returns exact value for $m = 1, 2, 3, 4, 5$. Does anyone know if there is an explicit formula for this definite integral for any positive integer $m$?
 [/INST] If you consider
$$I_m=2\pi \int_0^\infty \Big[J_m(x)\Big]^4\,dx$$ they are given in terms of Meijer G-functions.
Looking at the first
$$I_0=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
 1,1,1,1 \\
 \frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}
\end{array}
\right.\right) \qquad I_1=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
 0,1,1,2 \\
 \frac{1}{2},\frac{3}{2},-\frac{1}{2},\frac{1}{2}
\end{array}
\right.\right)\qquad I_2=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
 -1,1,1,3 \\
 \frac{1}{2},\frac{5}{2},-\frac{3}{2},\frac{1}{2}
\end{array}
\right.\right)$$
$$I_3=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
 -2,1,1,4 \\
 \frac{1}{2},\frac{7}{2},-\frac{5}{2},\frac{1}{2}
\end{array}
\right.\right) \qquad I_4=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
 -3,1,1,5 \\
 \frac{1}{2},\frac{9}{2},-\frac{7}{2},\frac{1}{2}
\end{array}
\right.\right)\qquad I_5=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
 -4,1,1,6 \\
 \frac{1}{2},\frac{11}{2},-\frac{9}{2},\frac{1}{2}
\end{array}
\right.\right)$$ and we may conjecture that
$$\color{blue}{I_m=G_{4,4}^{2,2}\left(1\left|
\begin{array}{c}
 1-m,1,1,m+1 \\
 \frac{1}{2},\frac{2m+1}{2},-\frac{2m-1}{2},\frac{1}{2}
\end{array}
\right.\right)}$$
This has been verified for much higher values of $m$ and the results compared to the results of numerical integration.
</s>