Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in [Formula: see text] modulo [Formula: see text]. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo [Formula: see text] are [Formula: see text]-automatic.

中文翻译：

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关于由 Thue-Morse 和倍周期 Stieltjes 连分数定义的序列的自动性

在过去的几十年里，自动数的连续分数展开得到了广泛的研究。研究兴趣一方面是在 1976 年 Baum 和 Sweet 的开创性论文之后的部分商的程度或自动性，另一方面是计算 Hankel 行列式和非理性指数，正如我们可以在Allouche-Peyrière-Wen-Wen、Bugeaud 和第一作者的作品。这篇论文的动机是逆问题：研究 Stieltjes 连分数，其系数形成一个自动序列。我们分别考虑由 Thue-Morse 和倍周期序列定义的两个这样的连分数，并证明它们与 [公式：参见文本] 模 [公式：参见文本] 中的代数级数一致。所以，