Transcendence of binomial and Lucas' formal power series

The formal power series (n greater than or equal to 0)Sigma((2n)(n))X-t(n) is transcendental over Q(X) when t is an integer greater than or equal to 2 . This is due to Stanley for t even, and independently to Flajolet and to W oodcock and Sharif for the general case. While Stanley and Flajolet used an alytic methods and studied the asymptotics of the coefficients of this seri es, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series module prime numbers p, and to use the p-Lucas proper ty: if n = Sigma n(i)p(i) is the base p expansion of the integer n, then ((2n)(n)) = Pi((2ni)(ni)) mod p. The series reduced module p is then proved algebraic over F-p(X), the field of rational functions over the Galois field F-p, but its degree is not a b ounded function of p. We generalize this method to characterize all formal power series that have the p-lucas property for "many" prime numbers p, and that are furthermore algebraic over Q(X). (C) 1998 Academic Press.