Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation

We develop exact expressions for the coefficients of series representations of trans-lations and rotations of local and multipole fundamental solutions of the Helmholtz equation in spherical coordinates. These expressions are based on the derivation of recurrence relations, some of which, to our knowledge, are presented here for the first time. The symmetry and other properties of the coefficients are also examined and, based on these, efficient procedures for calculating them are presented. Our expressions are direct and do not use the Clebsch–Gordan coefficients or the Wigner 3-j symbols, although we compare our results with methods that use these to prove their accuracy. For evaluating an Nt term truncation of the translated series (involving O(N2t) multipoles), our expressions require O(N3t) evaluations, compared to previous exact expressions that require O(N5t) operations.