Translational addition theorems for spheroidal scalar and vector wave functions

The translational addition theorems for spheroidal scalar wave functions $R_{mn}^{\left ( i \right )}\left ( {h, \xi } \right ){S_{mn}}\left ( {h, \eta } \right )\exp \left ( {jm\phi } \right ); i = 1, 3, 4$ and spheroidal vector wave functions $M_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \\ \eta , \phi } \right ), N_{mn}^{x, y, z\left ( i \right )}\left ( {h; \xi , \eta , \phi } \right ); i = 1, 3, 4$, with reference to the spheroidal coordinate system at the origin $O$, have been obtained in terms of spheroidal scalar and vector wave functions with reference to the translated spheroidal coordinate system at the origin $O’$, where $O’$ has the spherical coordinates (${r_0},{\theta _0},{\phi _0}$) with respect to $O$. These addition theorems are useful in acoustics and electromagnetics in those cases involving spheroidal radiators and scatterers.