Explicit evaluations of the Hankel determinants of a Thue--Morse-like sequence

Abstract

We obtain the explicit evaluations of the Hankel determinants of the formal power series ${\prod }_{k\ge 0}(1+J{x}^{3^k})$ where $J=(\sqrt{-3}-1)/2$, and prove that the sequence of Hankel determinants is an aperiodic automatic sequence taking value in $\{0,\pm 1,\pm J,\pm J^2\}$. This research is essentially inspired by the works about Hankel determinants of Thue--Morse-like sequences by Allouche, Peyrière, Wen and Wen (1998), Bacher (2006) and the first author (2013).