Summary: Note on the transcendence of a generating function
B“atiment 490, F91405 Orsay Cedex
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  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 .
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