Explicit evaluations of the Hankel determinants of a Thue--Morse-like sequence (1406.1594v1)

Abstract: We obtain the explicit evaluations of the Hankel determinants of the formal power series $\prod_{k\geq 0}(1+Jx^{3^{k}})$ 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, Peyri`ere, Wen and Wen (1998), Bacher (2006) and the first author (2013).