A new approach to the 2-regularity of the -abelian complexity of 2-automatic sequences

We prove that a sequence satisfying a certain symmetry property is 2-regular in the sense of Allouche and Shallit, i.e., the Z-module generated by its 2-kernel is finitely generated. We apply this theorem to develop a general approach for studying the-abelian complexity of 2-automatic sequences. In particular, we prove that the period-doubling word and the Thue–Morse word have 2-abelian complexity sequences that are 2-regular. Along the way, we also prove that the 2-block codings of these two words have 1-abelian complexity sequences that are 2-regular.