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

TL;DR

This paper explicitly evaluates the Hankel determinants of a specific Thue--Morse-like sequence, revealing its aperiodic automatic nature and value set, extending prior research on similar sequences.

Contribution

It provides explicit formulas for Hankel determinants of a new Thue--Morse-like sequence and proves its aperiodic automatic sequence properties.

Findings

Hankel determinants are explicitly evaluated.

The sequence is an aperiodic automatic sequence.

Values are in the set {0, ±1, ±J, ±J^2}.

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).