 
Summary: Transcendence of formal power series with rational
coefficients
J.P. Allouche
CNRS, LRI
Universit'e ParisSud, B“atiment 490
F91405 Orsay Cedex (France)
allouche@lri.fr
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,
pLucas sequences, ChomskySch¨utzenberger 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
n–0 a n X n (this series belongs to Z[[X]] ae Q[[X]]).
