Infinite Algebraic Independence of Polyadic Series with Periodic Coefficients

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