Linear independence of values of G-functions

Given any non-polynomial G-function F(z) = Sigma(infinity)(k=0)A(k)z(k) of radius of convergence R, we consider the G-functions F-n([s])(z) = Sigma(infinity)(k=0) A(k)/(k+n)(s)z(k+n) for any integers s >= 0 and n >= 1. For any fixed algebraic number a such that 0 < vertical bar alpha vertical bar < R and any number field K containing a and the A(k)'s, we define Phi(alpha,S) as the K-vector space generated by the values F-n([s]) (alpha) for n >= 1 and 0 < s < S. We prove that u(K) ,F log(S) <= dim(K)(Phi(s,S)) < v(F)S for any S, with effective constants u(K), F > 0 and v(F) > 0, and the family (F-n([s]) (alpha))(1 <= n <= vf, s >= 0 )contains infinitely many irrational numbers. This theorem applies in particular when F is a hypergeometric series with rational parameters or a multiple polylogarithm, and it encompasses a previous result by the second author and Marcovecchio in the case of polylogarithms. The proof relies on an explicit construction of Pade-type approximants. It makes use of results of Andre, Chudnovsky and Katz on G-operators, of a new linear independence criterion a la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.