  Addition theorems are principal tools that express a function $f (\bm{r} \pm
\bm{r}')$ in terms of products of other functions that only depend on either
$\bm{r}$ or $\bm{r}'$. The best known example of such an addition theorem is
the Laplace expansion of the Coulomb potential which possesses a characteristic
two-range form. Guseinov [Chem. Phys. {\bf 309}, 209 - 213 (2005)] derived
one-range addition theorems for the Coulomb potential via the limit $\beta \to
0$ in previously derived one-range addition theorems for the Yukawa potential
$\exp \bigl(-\beta | \bm{r}-\bm{r}'| \bigr) /| \bm{r}-\bm{r}'|$. At first
sight, this looks like a remarkable achievement, but from a mathematical point
of view, Guseinov's work is at best questionable and in some cases
fundamentally flawed. One-range addition theorems are expansions in terms of
functions that are complete and orthonormal in a given Hilbert space, but
Guseinov replaced the complete and orthonormal functions by nonorthogonal
Slater-type functions and rearranged the resulting expansions. This is a
dangerous operation whose validity must be checked. It is shown that the
one-center limit $\bm{r}' = \bm{0}$ of Guseinov's rearranged Yukawa addition
theorems as well as of several other addition theorems does not exist.
Moreover, the Coulomb potential does not belong to any of the Hilbert spaces
implicitly used by Guseinov. Accordingly, one-range addition theorems for the
Coulomb potential diverge in the mean. Instead, these one-range addition
theorems have to interpreted as expansions of generalized functions in the
sense of Schwartz that converge weakly in suitable functionals.
