On generalized Thue-Morse functions and their values

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This paper naturally extends and generalizes our previous work "Thue-Morse constant is not badly approximable", arXiv:1407.3182 [math.NT]. Here we consider the Laurent series fd(x) = ∏ₙ₌₀∞ (1 - x⁻ᵈⁿ), d, d≥ 2 which generalize the generating function f₂(x) of the Thue-Morse number, and study their continued fraction expansion. In particular, we show that the convergents of x⁻ᵈ⁺¹fd(x) have quite a regular structure. We address as well the question whether the corresponding Mahler numbers fd(a), a,d, a,d≥ 2, are badly approximable.