Integral of powers of Bessel function from 0 to infinity

Question

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 $m$?

Answer 1

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.