On the linear independence of values of \(G\)-functions (Q2212635)

Let \(\mathbb K\) be a number field and let \(F(z)=\sum_{k=0}^\infty A_kz^k\in\mathbb K[\![ z]\!]\) be a non-polynomial \(G\)-function. Let \(L\in\overline{\mathbb Q} [z,d/dz]\setminus \{ 0\}\) be an operator such that \(L(F(z))=0\) and of minimal order \(\mu\) for \(F\). Let \(\beta\in\mathbb Q\setminus\mathbb Z_{\leq 0}\). For positive integer \(n\) and non-negative integer \(s\) set \(F^{[s]}_{\beta ,n}(z)=\sum_{k=0}^\infty \frac{A_k}{(k+\beta +n)^s}\). Then, the author proves that for \(S\) large enough we have \[\frac{1+o(1)}{[\mathbb K:\mathbb Q]C(F,\beta)}\log S\leq \dim_{\mathbb K}\mathrm{Span}_{\mathbb K}(F^{[s]}_{\beta ,n}(\alpha);n\in\mathbb Z^+,0\leq s\leq S) \leq l_0(\beta)S+\mu . \] Here, \(\alpha\in\mathbb K\) with \(0\leq\mid\alpha\mid\leq R\) where \(R\) is the radius of convergence of \(F\) and \(l_0(\beta)\) and \(C(F,\beta)\) are constants.