Extension of some theorems of W. Schwarz

Title: Extension of some theorems of W. Schwarz
Creator: Coons, Michael
Relation: Canadian Mathematical Bulletin Vol. 55, p. 60-66
Publisher Link: http://dx.doi.org/10.4153/CMB-2011-037-9
Publisher: University of Toronto Press
Resource Type: journal article
Date: 2011
Description: 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.
Subject: functional equations; transcendence; power series
Identifier: http://hdl.handle.net/1959.13/1356383
Identifier: uon:31692
Identifier: ISSN:0008-4395
Language: eng
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