On the automaticity of sequences defined by continued fractions

Abstract

Continued fraction expansions of automatic sequences have been extensively studied during the last 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 exponent, as one can find in the works of Allouche-Peyrière-Wen-Wen, Bugeaud, and the first author. The present paper is motivated by the converse problem: to study 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 sequences of the coefficients of the power series expansions of the two continued fractions modulo 4 are 2-automatic. Our approach is to first guess the explicit formulas of certain subsequences of (Pn(x)) and (Qn(x)), where Pn(x)/Qn(x) is the canonical representation of the truncated continued fractions, then prove these formulas by an intricate induction involving eight subsequences while exploiting the relations between these subsequences.

On the algebraicity of Thue-Morse and period-doubling continued fractions

“…To our knowledge, little has been done in this direction. The authors [10] proved that the Stieltjes continued fractions defined by the Thue-Morse sequence and the perioddoubling sequence in Z [[x]] are congruent, modulo 4, to algebraic series in Z [[x]]. In 2020, Wu [16] obtained similar results concerning the Stieltjes continued fractions defined by the paperfolding sequence and the Golay-Shaprio-Rudin sequence.…”

“…15 [13, 17] 1 [3,4] 6 [9,4] 11 [8,13] 16 [18, 4] 2 [5,6] 7 [10,4] 12 [14,4] 17 [19,12] 3 [1,7] 8 [11,6] 13 [15,16] 18 [16,6] 4 [4,4] 9 [6,12] 14 [12,16] 19 [17,8] and output function n → τ (n): For all k ≥ 1, a 2 2k−1 = a + 1. Therefore we only have to check that identity (3.7) holds for k = 3, 5, 7, which turns out to be true.…”

“…As with real numbers, a formal Laurent series can also be represented by a continued fraction whose partial quotients are polynomials. Unlike for real numbers, the continued fraction expansion of an algebraic Laurent series of degree at least 3 may or may not have automatic partial quotients [4,5,14,2,15,11,12,13]; see also the introduction of [10].…”

Stieltjes continued fractions related to the paperfolding sequence and Rudin-Shapiro sequence

“…Our results are the following. Based on the result in [18] and our results, it is natural to ask that if one can characterize those automatic sequences a on the alphabet {−1, 1} such that Stiel a (x) is congruent modulo 4 to an algebraic series in Z…”

“…In [18], Han and Hu proved that the Stieltjes continued fractions given by the Thue-Morse sequence and the period doubling sequence are congruent modulo 4 to algebraic series in Z [[x]]. Here, we investigate the paperfolding sequence p and Rudin-Shapiro sequence r. We show that the Stieltjes continued fractions Stiel p (x) and Stiel r (x) are both congruent modulo 4 to algebraic series in Z [[x]].…”