The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields

We present simple alternate derivations of the translation formula for spherical vector multipole fields and the recurrence relations satisfied by the translation coefficients of spherical scalar and vector multipole fields. The derivations use the well-known and widely used spherical tensor technique to conveniently couple and decouple quantities associated with spherical angular variables, and these coupling and decoupling of angular quantities substantially simplify the algebra involved. As a consequence, the derivations are quite concise, and the resulting expressions for the recurrence relations are more compact and general than the existing ones.