The translational addition theorems for the spherical scalar and vector wave functions are derived in a novel, unified way based on the simple and well-known concepts of the radiation and incoming wave patterns. This approach makes the derivation simpler and more transparent compared to the previous approaches. As a result, we also obtain alternative and partly simpler expressions for the translation coefficients in the vector case.
Dufva, T., Sarvas, J., & Sten, J. (2008). Unified derivation of the translational addition theorems for the spherical scalar and vector wave functions. Progress In Electromagnetics Research B, 4, 79 - 99. https://doi.org/10.2528/PIERB07121203