Linear independence of values of $E$-functions with periodic coefficients

We consider sets of integers $a_{n}^{(k,j)}, j=1,...,m, k=1,...,T_{j}$ which satisfy conditions
$$a_{n}^{(k,j)}=a_{n+T_{j}}^{(k,j)} ,j=1,...,m, k=1,...,T_{j}, n=0,1,... $$
and functions
$$F_{j,k}(z)=\sum_{n=0}^{\infty}\frac{a_{n}^{(k,j)}}{n!}z^{n}, j=1,...,m, k=1,...,T_{j}. $$
We find conditions under which the set of functions
$$ 1, e^{z}, F_{j,k}(z), j=1,...,m, k=2,...,T_{j} $$
is linearly independent over $ \mathbb{C} (z) $ and for any rational $ \gamma\neq 0 $ their values at $\gamma$ are linearly independent numbers.An estimate of the measure of linear independence of these numbers is obtained. The result can be used to generate pseudo-random numbers.