Hankel Matrices for the Period-Doubling Sequence

TL;DR

This paper explicitly evaluates the Hankel determinants of the period-doubling sequence using Jacobsthal numbers, providing eigenvalues and eigenvectors of the associated matrices, and extends these results to other orders.

Contribution

It offers a novel explicit evaluation of Hankel determinants for the period-doubling sequence in terms of Jacobsthal numbers, including eigenvalues and eigenvectors.

Findings

Hankel determinants expressed via Jacobsthal numbers

Eigenvalues and eigenvectors of Hankel matrices derived

Results extended to other matrix orders

Abstract

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.