Non-commutative Rational Power Series and Algebraic Generating Functions

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 coe cients of 2-variable rational generating functions (Furstenberg's theorem). Algebraicity is of of practical as well as theoretical interest, since it guarantees an e cient recurrence for computing coe cients. Using now-classic results of Sch utzenberger on formal languages we prove: Theorem. Let K be a eld and f(X1; : : : ; Xk; Y1; : : : ; Yk) a rational power series in noncommmuting indeterminates. Then any coe cient of f(X1; : : : ; Xk; X 1 1 ; : : : ; X 1 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.