Abstract
AbstractIn this paper, we prove that a non–zero power series F(z) ∈ ℂ[[z]] satisfyingwhere 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 is transcendental for all algebraic numbers z with |z| < 1. We give a similar result for . 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.
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