Infinite algebraic independence of polyadic series with periodic coefficients

Consider sequences of integers $a^{(k,j)}_n$, $k=1,\dots, T_j$, $j=1,\dots,m$ such that $a^{(k,j)}_n=a^{(k,j)}_{n+T_j}$, $j=1,\dots,m$, $k=1,\dots,T_j$, $n=0,1,\dots$, and consider the series $F_{j,k}(z)=\sum_{n=0}^\infty a^{(k,j)}_n n! z^n$, $k=1,\dots,T_j$, $j=1,\dots,m$. The conditions are established under which the set of series $F_{j,k}(z)$, $k=2,\dots,T_j$, $j=1,\dots,m$ and the Euler series $\Phi(z)=\sum_{n=0}^\infty n!z^n$ are algebraically independent over $\mathbb C(z)$ and for any algebraic integer $\gamma\neq0$, their values at the point $\gamma$ are infinitely algebraically independent.