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Note on the transcendence of a generating function JeanPaul Allouche

Summary: Note on the transcendence of a generating function
Jean­Paul Allouche
B“atiment 490, F­91405 Orsay Cedex
I. Introduction
A word, i.e., a finite string of symbols taken from a finite set (sometimes called an
alphabet) is called periodic if it is the concatenation of identical words. For example
the word 001001 on the alphabet f0; 1g is periodic as it is the concatenation of two
copies of the word 001. A non­periodic word is called primitive. For example the
word 001101 on the alphabet f0; 1g is primitive.
In a recent paper [9] it is shown that the language of all primitive words on a
given finite alphabet cannot be unambiguous context­free, i.e., that the set of all the
primitive words on a finite set cannot be generated by a non­ambiguous grammar.
If we concentrate on the number­theoretical part of this result, we see that the main
tool is the following theorem [2].
Theorem. Let L be a language on the alphabet A. For each integer n, call u(n) the
number of words of L of length n. If the language L is unambiguous context­free,
then, the formal power series


Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11


Collections: Computer Technologies and Information Sciences