Abstract

Sequences of numbers abound in combinatorics whose generating functions are algebraic over the rational functions. Examples include Catalan and related numbers, numbers of words expressing an element in a free group, and diagonal coefficients of 2-variable rational generating functions (Furstenberg's theorem). Algebraicity is of of practical as well as theoretical interest, since it guarantees an efficient recurrence for computing coefficients. Using now-classic results of Schutzenberger on formal languages we prove: Theorem. Let K be a field and f(X 1 ; : : : ; X k ; Y 1 ; : : : ; Y k ) a rational power series in noncommmuting indeterminates. Then any coefficient of f(X 1 ; : : : ; X k ; X k ) converging w.r.t a given valuation on K is algebraic over K. Many algebraic generating functions, including those mentioned above are so as a consequence of this theorem; in particular it gives a new elementary proof of Furstenberg's theorem.

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