Multipole Matrix Elements of the Translation Operator

Formulas are given for the expansion of multipole fields of arbitrary tensorial character into multipole fields about a shifted origin. The expansion coefficients are given as matrix elements of the translation operator. In analogy to the matrix elements of the rotation operator, we introduce for these matrix elements a standard form which represents a parallel displacement of the coordinate system along the z axis. Any arbitrary translation of the coordinate system then consists of a consecutive application of a rotation, a standard translation, and a rotation. Since the multipole fields form a complete set any arbitrary function can in principle be expressed in a shifted coordinate system by means of the given formulas. All mathematical derivations are given.