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

Abstract

Given any non-polynomial G-function $$F(z)=\sum _{k=0}^\infty A_kz^k$$ of radius of convergence R and in the kernel a G-operator $$L_F$$ , we consider the G-functions $$F_n^{[s]}(z)=\sum _{k=0}^\infty \frac{A_k}{(k+n)^s}z^k$$ for every integers $$s\ge 0$$ and $$n\ge 1$$ . These functions can be analytically continued to a domain $${\mathcal {D}}_F$$ star-shaped at 0 and containing the disk $$\{\vert z\vert <R\}$$ . Fix any $$\alpha \in {\mathcal {D}}_F \cap \overline{{\mathbb {Q}}}^*$$ , not a singularity of $$L_F$$ , and any number field $${\mathbb {K}}$$ containing $$\alpha $$ and the $$A_k$$ ’s. Let $$\Phi _{\alpha , S}$$ be the $${\mathbb {K}}$$ -vector space spanned by the values $$F_n^{[s]}(\alpha )$$ , $$n\ge 1$$ and $$0\le s\le S$$ . We prove that $$u_{{\mathbb {K}},F}\log (S)\le \dim _{\mathbb {K}}(\Phi _{\alpha , S })\le v_FS$$ for any S, for some constants $$u_{{\mathbb {K}},F}>0$$ and $$v_F>0$$ . This appears to be the first general Diophantine result for values of G-functions evaluated outside their disk of convergence. This theorem encompasses a previous result of the authors in [Linear independence of values of G-functions, J. Europ. Math. Soc. 22(5), 1531–1576 (2020)], where $$\alpha \in \overline{{\mathbb {Q}}}^*$$ was assumed to be such that $$\vert \alpha \vert <R$$ . Its proof relies on an explicit construction of a Pade approximation problem adapted to certain non-holomorphic functions associated to F, and it is quite different of that in the above mentioned paper. It makes use of results of Andre, Chudnovsky and Katz on G-operators, of a linear independence criterion a la Siegel over number fields, and of a far reaching generalization of Shidlovsky’s lemma built upon the approach of Bertrand–Beukers and Bertrand.