Linear independence of values of $G$-functions

Abstract

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.

Linear independence of values of G-functions, II: outside the disk of convergence

“…(1.4) below immediately implies that this dimension is bounded above by ℓ 1 S + µ for every S ≥ 0, where the quantities ℓ 1 and µ will be defined later in the paper. Both can be computed from L F and are independent of K. We observe that ℓ 1 S + µ can in fact be replaced by ℓ 0 S + µ (where ℓ 0 is defined in the introduction of [13] and is ≤ ℓ 1 ) and provided one uses the analytic continuation to D F of Identity (5.2) of [13] instead of (1.4).…”

“…To deduce Corollary 1 from Theorem 1, one proceeds as in [13] by applying the analytic continuation to D F of Identity (5.2) of [13], with the same parameters as in [13]. That Identity (5.2) is formally similar to (1.4); the main difference is that for some values of the parameters we cannot take ℓ 1 = 1 in (1.4).…”

“…By (1.2), when |z 0 | > 1 and z 0 / ∈ [1, ∞), Li s (z 0 ) is a Q-linear combination of Li s (1/z 0 ) (defined by the convergent series ∞ k=1 z −k 0 /k s ) and powers of log(z 0 ) and 2iπ. The proof of Theorem 1 is not a simple generalization of the corresponding theorem in [13], even though the strategy is similar at the beginning. There are important difficulties.…”

“…where, in particular, the parameter ℓ 1 is defined in §3.1. For |z| < R, in [13] we applied the saddle point method to estimate |J F (z)| precisely; this enabled us to apply a generalization of Nesterenko's linear independence criterion [20] and deduce Theorem 1 in this case. When |z| ≥ R, the first thing we would like to point out is that |J F (z)| is still small enough to prove linear independence results; this could seem surprising since the series (1.3) is divergent if |z| > R. Instead, we use an integral representation of J f (z) to bound |J F (z)| from above (see §8.1).…”

“…In §2 we focus on the two main tools in our approach. First we extend in §2.1 the definition of f [s] j , given in [13] when f is holomorphic at 0, to any function in the Nilsson class with rational exponents at 0; then we study in §2.2 the variation around a singularity of a solution of a Fuchsian operator, which generalizes the weight of polylogarithms. Section 3 is devoted to the statement of Theorem 2, namely the Padé approximation problem satisfied by the polynomial coefficients of (1.4).…”