With the following notations:
- $j_n$: spherical Bessel functions,
- $y_n$: spherical Neumann function,
- $P_n$: Legendre polynomial,
- $r$, $\rho$, $\theta$, $\lambda$ arbitrary complex, $R=\sqrt{r^2+\rho^2-2r\rho\cos\theta},$
I want a justification of those two expansions: \begin{align} \frac{\sin(\lambda R)}{\lambda R}=\sum_{n=0}^\infty (2n+1)\,j_n(\lambda R)\,j_n(\lambda \rho)\,P_n(\cos\theta),\\ -\frac{\cos(\lambda R)}{\lambda R}=\sum_{n=0}^\infty (2n+1)\,j_n(\lambda R)\,y_n(\lambda \rho)\,P_n(\cos\theta) \end{align} with $|r e^{\pm i\theta}|<|\rho|$.