Let f be a G-function (in the sense of Siegel), and α be an algebraic number; assume that the value f (α) is a real number. As a special case of a more general result, we show that this number can be written as g (1), where g is a G-function with rational coefficients and arbitrarily large radius of convergence. As an application, we prove that quotients of such values are exactly the numbers which can be written as limits of sequences a n /b n , where ∞ n=0 a n z n and ∞ n=0 b n z n are G-functions with rational coefficients. This result provides a general setting for irrationality proofs in the style of Apéry for ζ(3), and gives answers to questions asked by T. Rivoal in [Approximations rationnelles des valeurs de la fonction Gamma aux rationnels : le cas des puissances, Acta Arith. 142 (2010), no. 4, 347-365].

Given any non-polynomial G-function F (z) = ∞ k=0 A k z k of radius of convergence R, we consider the G-functions FA k (k+n) s z k for any integers s ≥ 0 and n ≥ 1. For any fixed algebraic number α such that 0 < |α| < R and any number field K containing α and the A k 's, we define Φ α,S as the K-vector space generated by the values Ffor any S, with effective constants u K,F > 0 and v F > 0, and that the family F [s] n (α) 1≤n≤v F ,s≥0 contains infinitely many irrational numbers. This theorem applies in particular when F is an 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 Padé-type approximants. It makes use of results of André, Chudnovsky and Katz on G-operators, of a new linear independence criterionà la Nesterenko over number fields, of singularity analysis as well as of the saddle point method.

This text is devoted to simultaneous approximation to ξ and ξ 2 by rational numbers with the same denominator, where ξ is an irrational non-quadratic real number. We focus on an exponent β 0 (ξ) that measures the regularity of the sequence of all exceptionally precise such approximants. We prove that β 0 (ξ) takes the same set of values as a combinatorial quantity that measures the abundance of palindromic prefixes in an infinite word w. This allows us to give a precise exposition of Roy's palindromic prefix method. The main tools we use are Davenport-Schmidt's sequence of minimal points and Roy's bracket operation.In this Section, we recall the properties [7] of words with many palindromic prefixes, and also the bracket operation introduced by Roy [12].