Simple continued fractions for the Fredholm numbers

Let {a_i} with a₁ ≥ 2 be an infinite bounded sequence of positive integers, and d₁ = 1, d_i = ±1 for i = 2, 3,…. Let {Q_i} be another sequence defined by the recursion Q₁ = 1, Q_i = a_i?1Q_i?1^k for i = 2, 3,…, where k ≥ 2 an integer. Put C_k(a) = Σ_{i = 1}^∞d_iQ_i^?1. In this paper we shall determine the simple continued fraction expansion for the real numbers C_k(a).