On the linear independence of values of G-functions

We consider a G-function F(z)=∑k=0∞Akzk∈K〚z〛, where K is number field, of radius convergence R and annihilated by the G-operator L∈K(z)[d/dz], parameter β∈Q∖Z⩽0. define family G-functions Fβ,n[s](z)=∑k=0∞Ak(k+β+n)szk+n indexed integers s n. Fix α∈K⁎∩D(0,R). Let Φα,β,S be K-vector space generated values Fβ,n[s](α), n∈N, 0⩽s⩽S. show that there exist some positive constants uK,F,β vF,β such uK,F,βlog⁡(S)⩽dimK⁡Φα,β,S⩽vF,βS. This generalizes previous theorem Fischler Rivoal (2017), which case β=0. Our proof an adaptation their article [6], making use André-Chudnovsky-Katz Theorem on structure G-operators saddle point method. Moreover, we rely Dwork André's quantitative results size to obtain explicit formula for constant uK,F,β, was not given in [6]