Computing powers of two generalizations of the logarithm

We prove multiple-series representations for positive integer powers of the series L(z; α) = [formula could not be replicated], |z| < 1, α ≥ 0, and ℓ_q(z) = [formula could not be replicated], |z| ≤ 1, |q| < 1. The results generalize a known formula for powers of the series for the ordinary logarithm -log(1-z) = L(z;0).