On The Automaticity Of Sequences Defined By The Thue-Morse And Period-Doubling Stieltjes Continued Fractions

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-Peyriere-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 Z[[x]] modulo 4. Consequently, the sequence of the coefficients of the power series expansions of the two continued fractions modulo 4 are 2-automatic.