Algebraic independence of certain power series of algebraic numbers

AbstractLet f(z) = Σk = 0∞ zk!. Then in p-adic field we prove that for any algebraic numbers α1 ,…, αn with 0 < |αi| < 1 (1 ⩽ i ⩽ n), f(α1),…, f(αn) are algebraically independent over Q if and only if αiαj 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