I would like to carry out symbolically the following integral
$\int_0^\infty d r \,r^2\, j_0( k r)\, j_0( k_1 r)\, j_0( k_2 r)\,, $ where $j_0(r)$ is the zeroth order spherical Bessel function and $k$,$k_1$ and $k_2$ are real numbers.
Idea?
I am wondering if I should use this expansion
$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m}$
from this reference
Clue
If I am to believe Mathematica $\int_0^\infty d r \,r^2\, j_0( r)\, j_0( 2 r)\, j_0( 3 r)=\frac{\pi}{48}$ for instance, so the integral seems possible.