On the values of $G$-functions

Abstract

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].

Arithmetic theory of $E$-operators

“…The situation is much more complicated around ∞, which is in general an irregular singularity of L; this part is therefore much more involved than the corresponding one for G-functions [9] (since ∞ is a regular singularity of G-operators, the connexion constants of G-functions at any ζ ∈ C ∪ {∞} always belong to G). The local solutions at ∞ involve divergent series, which give rise to Stokes phenomenon: the expression of an E-function E(z) on a given basis is valid on certain angular sectors, and the connexion constants may change from one sector to another when crossing certain rays called anti-Stokes directions.…”

“…In [9] we have proved that a complex number α belongs to the fraction field Frac G of G if, and only if, there exist sequences (P n ) and (Q n ) of algebraic numbers such that lim n P n /Q n = α and n≥0 P n z n , n≥0 Q n z n are G-functions. We have introduced this notion in order to give a general framework for irrationality proofs of values of G-functions such as zeta values.…”

“…, then for any x, y ∈ Q \ Z ≤0 we have Γ(x)Γ(y) = Γ(x + y)B(x, y) ∈ S because B(x, y) ∈ G (see [9]). This can also be proved directly from the definition of S: for any x, y ∈ Q \ Z ≤0 , we have…”

“…If X is a ring then we denote by Frac X = X X its field of fractions. We recall [9] that B(x, y) belongs to the group of units G * of G for any x, y ∈ Q, so that Γ induces a group homomorphism Q → C * /G * (by letting Γ(−x) = 1 for x ∈ N). Therefore Γ(Q) · G * is a subgroup of C * , and so is Γ(Q) · exp(Q) · Frac G; for future reference we write…”

“…In this paper we consider the following set E, which is analogous to the ring G of values at algebraic points of analytic continuations of G-functions studied in [9]; we recall that G might be equal to P [1/π], where P is the ring of periods (in the sense of Kontsevich-Zagier [13]: see §2.2 of [9]). …”