Abstract. In this paper, we prove that for every integer k ≥ 1, the k-abelian complexity function of the Cantor sequence c = 101000101 · · · is a 3-regular sequence.

In this paper, we study the abelian complexity of the Rudin-Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function ρ(n) which satisfies certain recurrence relations. As a consequence, the abelian complexity function is 2-regular. Further, we prove that the box dimension of the graph of the asymptotic function λ(x) is 3/2 where λ(x) = lim k→∞ ρ(4 k x)/ √ 4 k x and ρ(x) = ρ( x ) for any x > 0.

In this paper, we study the 2-binomial complexity b tm,2 (n) of the generalized Thue-Morse words t m for every integer m ≥ 3. We obtain the exact value of b tm,2 (n) for every integer n ≥ m 2 . As a consequence, b tm,2 (n) is ultimately periodic with period m 2 . This result partially answers a question of M. Lejeune, J. Leroy and M. Rigo [Computing the k-binomial complexity of the Thue-Morse word, J. Comb. Theory Ser. A, 176 (2020) 105284].

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