Arithmetical properties of a certain power series

The function f(θ, φ; x, y) = Σ_{k = 1}^∞ Σ_{1 ≤ m ≤ kθ + φ} x^ky^m, where θ > 0 is irrational and φ is real, satisfies Mahler-type functional equations which enable us to represent it by a gap-like series and then by a continued fraction. Using these representations, we describe the sequence {[(k + 1) θ + φ] - [kθ + φ]}_{k = 1}^∞ by a chain of substitutions and give algebraic independence results for the values of f(θ, φ, x, y) at some algebraic points when the partial quotients of the continued fraction of θ are unbounded, and irrationality measures for the values at some rational points.