Algebraic independence over ℚ_p

Let f(x) be a power series ∑_n≥1 ζ(n)x^e(n), where (e(n)) is a strictly increasing linear recurrence sequence of nonnegative integers, and (ζ(n)) a sequence of roots of unity in ℚ_p satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over ℚ_p of the elements f(α₁),…, f(α_t) from ℂ_p in terms of the distinct α₁,…, α_t ∈ ℚ_p satisfying 0 < |α_τ |p < 1 for τ = 1,…, t. A striking application of our basic result says that, in the case e(n) = n, the set {f(α)| α ∈ ℚ_p, 0 < |α|_p < 1} is algebraically independent over ℚ_p if (ζ(n)) satisfies the “technical condition”. We close with a conjecture concerning more general sequences (e(n)).