Linear independence of values of some hypergeometric functions (Q1074631)

Let us define \[ F(\alpha +1,1,\gamma +1;z)=\sum^{\infty}_{r=0}\frac{(\alpha +1)...(\alpha +r)}{(\gamma +1)...(\quad \gamma +r)}z^ r,\quad \alpha,\gamma \in {\mathbb{Q}},\quad \gamma \neq -1,-2,.... \] Further, let \(f_ i(z)=F(\alpha +1,1,\gamma +1;z_ iz)\), \(i=1,...,s\), where \(z_ 1,...,z_ s\) are non-zero rational numbers satisfying \(z_ i\neq z_ j\), \(\forall i\neq j\). Using the results of \textit{Yu. V. Nesterenko} [Vestn. Mosk. Univ., Ser. I 1985, No.1, 46-49 (1985; Zbl 0572.10027)] the author obtains sharp explicit estimates for linear forms \[ | x_ 0+x_ 1 f_ 1(z)+...+x_ s f_ s(z)|,\quad x_ i\in {\mathbb{Z}}, \] at certain rational points z.