Infinite Algebraic Independence of Polyadic Series with Periodic Coefficients

Abstract

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.