Also, we use the following result (Epton and Dembart[6]):

where the inequality is assumed to hold. In Eq. (8) stands for the unit vector given by

where and stand for the spherical Hankel function and the Legendre function, respectively.

We now compute the integral on the RHS of Eq. (5) over a subset of
denoted by for an which is away from . Using
Eqs. (7) and (8) we obtain

where and are the coefficients of local expansions centred at expressed in terms of the multipole moments and centred at as follows:

The multipole moments and are given by

Eqs. (10) and (11) are the M2L translation formulae which dominate the performance of FM-BIEM. Note that Eqs. (10) and (11), written for many s, have the form of the product of a diagonal matrix and a vector, in contrast to the M2L translation formulae in FM-BIEM with the Wigner-3j symbols which have the form of the product of a full matrix and a vector. Also, we notice that this formulation is in terms of 4 multipole moments (3 for and 1 for ) and 4 coefficients of the local expansion (3 for and 1 for ).