Extension of Some Theorems of W. Schwarz
2012 ◽
Vol 55
(1)
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pp. 60-66
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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.
2008 ◽
Vol 144
(1)
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pp. 119-144
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1966 ◽
Vol 62
(4)
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pp. 637-642
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1968 ◽
Vol 9
(2)
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pp. 146-151
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1959 ◽
Vol 55
(1)
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pp. 51-61
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1966 ◽
Vol 9
(05)
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pp. 757-801
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1982 ◽
Vol 34
(4)
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pp. 952-960
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1970 ◽
Vol 13
(1)
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pp. 151-152
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1978 ◽
Vol 26
(1)
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pp. 31-45
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1982 ◽
Vol 33
(2)
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pp. 171-178
1990 ◽
Vol 33
(3)
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pp. 483-490
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