Hankel Determinant Calculus for the Thue-Morse and related sequences

TL;DR

This paper introduces a novel method for evaluating Hankel determinants of automatic sequences, including Thue-Morse, using modular arithmetic and sequence transformations, simplifying proofs of non-vanishing properties.

Contribution

The paper presents a new approach to compute Hankel determinants of automatic sequences via modular equivalences, enabling closed-form expressions and simplified proofs.

Findings

Hankel determinants of the Thue-Morse sequence are nonzero.

The method simplifies existing proofs of non-vanishing determinants.

Hankel determinants do not vanish when powers are replaced by 3^n.

Abstract

The Hankel determinants of certain automatic sequences $f$ are evaluated, based on a calculation modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have any closed-form expressions; the traditional methods, such as $LU$-decompo\-si\-tion and Jacobi continued fraction, cannot be applied directly. Our method is based on a simple idea: the Hankel determinants of each sequence $g$ equal to $f$ modulo $p$ are equal to the Hankel determinants of $f$ modulo $p$. The clue then consists of finding a nice sequence $g$, whose Hankel determinants have closed-form expressions. Several examples are presented, including a result saying that the Hankel determinants of the Thue-Morse sequence are nonzero, first proved by Allouche, Peyri\`ere, Wen and Wen using determinant manipulation. The present approach shortens the proof of the latter result significantly. We also prove that the corresponding Hankel determinants do not vanish when the powers $2^n$ in the infinite product defining the $\pm 1$ Thue--Morse sequence are replaced by $3^n$.