On the linear independence of values of G-functions

Abstract

We consider a G-function F (z) = ∞ k=0 A k z k ∈ K z , where K is a number field, of radius of convergence R and annihilated by the G-operator L ∈ K(z) [d/dz], and a parameter β ∈ Q \ Z 0 . We define a family of G-functions F [s] β,n (z) = ∞ k=0 A k (k + β + n) s z k+n indexed by the integers s and n. Fix α ∈ K * ∩ D(0, R). Let Φ α,β,S be the K-vector space generated by the values F [s] β,n (α), n ∈ N, 0 s S. We show that there exist some positive constants u K,F,β and v F,β such that u K,F,β log(S) dim K Φ α,β,S v F,β S. This generalizes a previous theorem of Fischler and Rivoal (2017), which is the case β = 0. Our proof is an adaptation of their article [6], making use of the André-Chudnovsky-Katz Theorem on the structure of the G-operators and of the saddle point method. Moreover, we rely on Dwork and André's quantitative results on the size of G-operators to obtain an explicit formula for the constant u K,F,β , which was not given in [6] in the case β = 0. This family of special functions has been studied together with the family of E -functions, which are the functions f (z) = ∞ k=0 a k k! z k satisfying a) and such that the a k satisfy the conditions b) and c), since Siegel defined them in 1929 [15]. The most basic example of G-function, which gives it its name, is the geometric series f(−1) k z 2k+1 2k + 1 and the family of hypergeometric functions with rational parameters: if α = (α 1 , .

Quantitative problems on the size of G-operators

“…Let us now consider a Diophantine approximation problem. It is a generalization of results by Fischler and Rivoal ([6]) studied in [11].…”

“…Thus, the combination of ( 10) and (11) shows that σ(L 4 ) 1 + log(2) max σ(L 1 ), σ(L 2 ) , so that L 4 is a G-operator.…”

“…The first one consists, given two G-operators L 1 and L 2 , in finding an explicit inequality between the size of L 1 L 2 and the sizes of L 1 and L 2 . Using the first application and Theorem 2, we then give an application consisting in the evaluation of a constant appearing in a Diophantine problem, which is studied in another paper [11].…”

“…We now give two applications of the results of Sections 2 and 3. The first one consists in expressing the size of a product of G-operators in terms of the sizes of each term of the product; the second one is related to a Diophantine problem studied in [11].…”

“…Chudnovsky's Theorem was proved in such a way that there is an explicit relation between the size of G and the size of f (see [5,chapter VII]). This is particularly useful for Diophantine applications, as shown in [11].…”