Algebraic independence of certain power series of algebraic numbers

Let f(z) = Σ_{k = 0}^∞ z^k!. Then in p-adic field we prove that for any algebraic numbers α₁,..., α_n with 0 < |α_i| < 1 (1 ≤ i ≤ n), f(α₁),..., f(α_n) are algebraically independent over Q if and only if α_i α_j is not a root of unity for i ≠ j. In the complex field we prove the above result only when n = 2, making use of the p-adic field.