- 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 Gk(z) := Σ∞n=0 zkn (1 − zkn)−1 is transcendental for all algebraic numbers z with |z| < 1. We give a similar result for Fk(z) := Σ∞n=0 zkn)(1 + zkn)−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
- Full Text
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