Analytic Models and Ambiguity of Context-Free Languages

We establish that several classical context-free languages are inherently ambiguous by proving that their counting generating functions, when considered as analytic functions, exhibit some characteristic form of transcendental behaviour. To that purpose, we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages. In particular, the paper contains a general density theorem for unambiguous context-free languages.