Physics Asked by Boa_Constrictor on May 8, 2021

Is there a similar version of the summation theorem for spherical Bessel functions $j_{k}$ that of standard Bessel function $J_{k}$? Especially look below Eq.(3) in this paper https://arxiv.org/abs/cond-mat/0510271 for $J_{0}left(left|p-p’right|rright)$.

$$J_{0}left(left|vec{p}-vec{p’}right|rright)=sum_{k=-infty}^{infty}J_{k}left(prright)J_{k}left(p’rright)cos ktheta$$

where $theta$ is the angle between the two vectors in the plane.

I am interested to know whether a similar expression exist for $j_{0}left(left|vec{p}-vec{p’}right|rright)$, where the vector lies in three dimension.

The identity you seek is

$$j_0(|vec{p}-vec{p},'|r)=sum_{n=0}^{infty}(2n+1)j_n(pr)j_n(p'r)P_n(costheta).$$

It follows from (10.60.2) here.

Answered by G. Smith on May 8, 2021

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