Bases formed of successive primitives

Necessary and sufficient conditions are found in order for the system of successive primitives
$$ \biggl\{F_n(z)=\sum_{k=0}^\infty\frac{a+_{k-n}}{k!}z^k\biggr\},\quad n=0,1,2,\dots, $$
generated by the integer-valued function $F_0(z)=\sum_{k=0}^\infty\frac{a_{k_{zk}}}{k!}$ growth no higher than first order of the normal type $\sigma(F_0(z)\in[1,\sigma]$, to form a quasi-power basis in the class $[1;\sigma]$.