Arithmetic properties of the values of a set of functions satisfying a system of differential equations

General results were presented in [2] and [3] concerning arithmetic properties of the values at algebraic points of a class of analytic functions satisfying linear differential equations.
In the present note we consider the application of these results to the set of functions
$$ f(\alpha_kz)=\sum_{n=0}^\infty\frac{\mu(\mu+1)\dots(\mu+n-1)}{\lambda(\lambda+1)\dots(\lambda+n-1)}(\alpha_kz)^n(k=1,2,\dots,m,\quad\lambda\ne0,-1,-2,\dots), $$
where $\alpha_1,\dots,\alpha_n$ are algebraic numbers; $\lambda$ and $\mu$ are rational numbers; and the functions satisfy a system of linear differential equations.