Extension of some theorems of W. Schwarz

In this paper, we prove that a non–zero power series F(z) ∈ C[[z]] satisfying [formula could not be replicated] where d ≥ 2, A(z), B(z) ∈ C[z] with A(z) ≠ 0 and deg A(z), deg B(z) < d is transcendental over 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 ≥ 2 the series G_k(z) := Σ^∞_n=0 z^kn (1 − z^kn)⁻¹ is transcendental for all algebraic numbers z with |z| < 1. We give a similar result for F_k(z) := Σ^∞n=0 z^kn)(1 + z^kn)⁻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.