Extension of Some Theorems of W. Schwarz

Coons, Michael

Canadian mathematical bulletin, Tome 55 (2012) no. 1, pp. 60-66

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-037-9

Mots-clés : 11B37, 11J81, functional equations, transcendence, power series