Transcendence of binomial and Lucas' formal power series

[摘要] 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 Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, 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 property: 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 bounded 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.

[发布日期] 1998-12-15