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Transcendence of Formal Power Series With Rational Coefficients (1999)
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BibTeX
@MISC{Allouche99transcendenceof,
author = {J.-P. Allouche},
title = {Transcendence of Formal Power Series With Rational Coefficients},
year = {1999}
}
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Abstract
We give algebraic proofs of transcendence over Q(X) of formal power series with rational coefficients, by using inter alia reduction modulo prime numbers, and the Christol theorem. Applications to generating series of languages and combinatorial objects are given. Keywords: transcendental formal power series, binomial series, automatic sequences, p-Lucas sequences, Chomsky-Schutzenberger theorem. 1 Introduction Formal power series with integer coefficients often occur as generating series. Suppose that a set E contains exactly a n elements of "size" n for each integer n: the generating series of this set is the formal power series P n0 a n X n (this series belongs to Z[[X]] ae Q[[X]]). Properties of this formal power series reflect properties of its coefficients, and hence properties of the set E. Roughly speaking, algebraicity of the series over Q(X) means that the set has a strong structure. For example, the Chomsky-Schutzenberger theorem [16] asserts that the generating seri...
Keyphrases
formal power series rational coefficient chomsky-schutzenberger theorem binomial series strong structure hence property introduction formal power series roughly speaking automatic sequence combinatorial object p-lucas sequence formal power series reflect property integer coefficient transcendental formal power series algebraic proof christol theorem
