Hankel determinants of a Sturmian sequence

TL;DR

This paper characterizes a specific Sturmian sequence generated by a substitution, analyzes the distribution of zeros in its Hankel determinants, and provides explicit formulas for these determinants across all indices.

Contribution

It offers a new characterization of the Sturmian sequence via $f$-representation and explicitly computes its Hankel determinants for all indices.

Findings

Distribution of zeros induces a partition of integer lattices

Explicit formulas for Hankel determinants $H_{m,n}$

Characterization of the sequence using $f$-representation

Abstract

Let $\tau$ be the substitution $1\to 101$ and $0\to 1$ on the alphabet $\{0,1\}$. The fixed point of $\tau$ leading by 1, denoted by $\mathbf{s}$, is a Sturmian sequence. We first give a characterization of $\mathbf{s}$ using $f$-representation. Then we show that the distribution of zeros in the determinants induces a partition of integer lattices in the first quadrant. Combining those properties, we give the explicit values of the Hankel determinants $H_{m,n}$ of $\mathbf{s}$ for all $m\ge 0$ and $n\ge 1$.