Published online by Cambridge University Press: 20 November 2018
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.