Estimate of a linear form in the values of some functions

We establish effective lower bounds for a linear form in the values at distinct algebraic points of the functions
\begin{align} f_k(z)&=\sum_{n=1}^\infty\frac1{((n-1)!)^mn^k}\biggl(\frac{{}'z}m\biggr)^{mn}, \quad k=1,\dots,m-1,\notag\\ f_m=(z)&=\sum_{n=0}^\infty\frac1{(n!)^m}\biggl(\frac{z}m\biggr)^{mn}. \notag \end{align}