Algebraic independence of subseries over arithmetical progressions of certain exponential power series

Let q≥2q≥2 be an integer. We separate a given power series into qq subseries according to the residue classes mod qq of their powers. We study algebraic independence for values at an algebraic point of the qq subseries of certain exponential power series including the exponential generating function of Fibonacci numbers. Our proofs rely on Lindemann–Weierstrass theorem.