  Addition theorems can be constructed by doing three-dimensional Taylor
expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot
\mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in
addition theorems of irreducible spherical tensors, the application of the
translation operator in its Cartesian form $\exp (x' \partial /\partial x) \exp
(y' \partial /\partial y) \exp (z' \partial /\partial z)$ would lead to
enormous technical problems. A better alternative consists in using a series
expansion for the translation operator $\exp (\mathbf{r}' \cdot
\mathbf{\nabla})$ involving powers of the Laplacian $\mathbf{\nabla}^2$ and
spherical tensor gradient operators $\mathcal{Y}_{\ell}^{m} (\nabla)$, which
are irreducible spherical tensors of ranks zero and $\ell$, respectively [F.D.\
Santos, Nucl. Phys. A {\bf 212}, 341 (1973)]. In this way, it is indeed
possible to derive addition theorems by doing three-dimensional Taylor
expansions [E.J. Weniger, Int. J. Quantum Chem. {\bf 76}, 280 (2000)]. The
application of the translation operator in its spherical form is particularly
simple in the case of $B$ functions and leads to an addition theorem with a
comparatively compact structure. Since other exponentially decaying functions
like Slater-type functions, bound-state hydrogenic eigenfunctions, and other
functions based on generalized Laguerre polynomials can be expressed by simple
finite sums of $B$ functions, the addition theorems for these functions can be
written down immediately.
