I read in the appendix of a paper that, in the limit where $k_0 \ll k$, we can expand:
$$ j_{\ell}(|\mathbf{k}\pm \mathbf{k}_0|) Y_{\ell m}\left( \frac{\mathbf{k}\pm \mathbf{k}_0}{|\mathbf{k}\pm \mathbf{k}_0|} \right) \simeq j_{\ell}(k) Y_{\ell m}(\hat{k}) \mp R_{\ell m}^{1,\ell+1}j_{\ell+1}(k)Y_{\ell+1,m}(\hat{k})\pm R_{\ell m}^{1,\ell-1} j_{\ell-1}(k) Y_{\ell-1,m}(\hat{k}) $$ where $$ R_{\ell m}^{l_1 l_2} =(-1)^m \sqrt{ \frac{(2\ell+1)(2\ell_1+1)(2\ell_2+1)}{4 \pi}} \left( \begin{array}[ccc] e\ell_1 & \ell_2 & \ell \\ 0 & 0 & 0 \end{array}\right) \left( \begin{array}[ccc] e\ell_1 & \ell_2 & \ell \\ 0 & m & -m \end{array}\right). $$
The matrices are the 3j Wigner symbols, $j$ is the spherical Bessel function. Can someone explain how this expansion is obtained? Many thanks.
It is a Taylor expansion of the Bessel function and spherical harmonics. It can be derived by writing the few first terms of the expansion, and using relations for Bessel functions as well as recursion relation for the derivative of spherical harmonics.