I'm trying to find the first few terms in the spherical Bessel functions of the $1^{st}$ kind and am not getting the third term correct.
$j_l(kr)=\left(\frac{-r}{k}\right)^l\left(\frac{1}{r}\frac{d}{dr}\right)^lj_{0}(kr)$,
and
$j_{0}(kr)=sinc(kr)=\frac{sin(kr)}{kr}$.
If $k=1$, and $l=2$, I am getting the following:
$j_2(r)=r^2\left(\frac{1}{r^2}\right)\left(\frac{d^2}{dr^2}\left(\frac{sin(r)}{r}\right)\right)=-\frac{1}{r^3}\left((r^2-2)sin(r)+2rcos(r)\right)$.
But it should be
$j_2(r)=\frac{3sin(r)}{r^3}-\frac{3cos(r)}{r^2}-\frac{sin(r)}{r}$.
When I graph both solutions, they do not have the same roots. I am making a mistake but I can't see where...