Arithmetic properties of values of analytic functions connected by algebraic equations over fields of rational functions

In this paper theorems are proved about the arithmetic character of the values at algebraic points of a collection of $G$-functions which constitute a solution of a system of linear differential equations with coefficients from $C(z)$ connected by algebraic equations over $C(z)$. In addition, the theorems on E-functions proved by A. B. Shidlovskii in 1962 are supplemented.