In this paper, we prove that a non–zero power series
$F(z\text{)}\in \mathbb{C}\text{ }[[z]]$
satisfying
$$F({{z}^{d}})\,=\,F(z)\,+\,\frac{A(z)}{B(z)},$$
where
$d\,\ge \,2,\,A(z),\,B(z)\,\in \,\mathbb{C}[z]$
, with
$A(z)\,\ne \,0$
and
$\deg \,A(z),\,\deg \,B(z)\,<\,d$
is transcendental over
$\mathbb{C}(z)$
. Using this result and a theorem of Mahler’s, we extend results of Golomb and Schwarz on transcendental values of certain power series. In particular, we prove that for all
$k\,\ge \,2$
the series
${{G}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1-{{z}^{{{k}^{n}}}})}^{-1}}$
is transcendental for all algebraic numbers
$z$
with
$\left| z \right|\,<\,1$
. We give a similar result for
${{F}_{k}}(z):=\mathop{\sum }_{n=0}^{\infty }{{z}^{{{k}^{n}}}}{{(1+{{z}^{{{k}^{n}}}})}^{-1}}$
. These results were known to Mahler, though our proofs of the function transcendence are new and elementary; no linear algebra or differential calculus is used.