A succinct way to diagonalize the translation matrix in three dimensions

The diagonalization of the translation matrix is crucial in reducing the solution time in the fast multipole method. The translation matrix can be related, to the matrix representation of the translation operators in the translation group in group theory. Therefore, these matrices can be diagonalized with a proper choice of basis representation. Here, a different and succinct way to diagonalize the translation operator in three dimensions for the Helmholtz equation involving a general number of multipoles is demonstrated. The derivation is concise, and can be related to a set of similarity transforms equivalent to the change of basis representation for the translation group. The result can be used for scattering calculations related to the wave equation as found in electrodynamics, elastodynamics, and acoustics, where the fast multipole method is used.