INFINITE ALGEBRAIC INDEPENDENCE OF POLYADIC SERIES WITH PERIODIC COEFFICIENTS

Consider sequences of integers 𝑎𝑛(𝑘,𝑗), 𝑘 = 1, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚 such that 𝑎𝑛(𝑘,𝑗) = 𝑎𝑛(𝑘+,𝑇𝑗)𝑗 , 𝑗 = 1, … , 𝑚, 𝑘 = 1, … , 𝑇𝑗, 𝑛 = 0, 1, …, and consider the series 𝐹𝑗,𝑘(𝑧) = ∞ ∑ 𝑛=0 𝑎(𝑘,𝑗)𝑛 𝑛!𝑧𝑛, 𝑘 = 1, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚. The conditions are established under which the set of series 𝐹𝑗,𝑘(𝑧), 𝑘 = 2, … , 𝑇𝑗, 𝑗 = 1, … , 𝑚 and the Euler series Φ(𝑧) = ∞ ∑ 𝑛=0 𝑛!𝑧𝑛 are algebraically independent over ℂ(𝑧) and for any algebraic integer γ ≠ 0, their values at the point γ are infinitely algebraically independent.